PROFESSOR TODA: Any questions so far?
I mean, conceptual, theoretical questions first,
and then we will do the second part
of [INAUDIBLE] applications.
Then you can ask for more questions.
No questions so far?
I have not finished 11-4.
I still owe you a long explanation about 11-4.
Hopefully it's going to make more sense today
than it made last time.
I was just saying that I'm doing 11-4.
This is a lot of chapter.
So second part of 11-4 today-- tangent plane and applications.
Now, we don't say what those applications are
from the start, but these are some very important concepts
called the total differential.
And the linear approximation number
is going under the [INAUDIBLE].
Thank you, sir.
Linear approximation for functions of the type z
equals f of xy, which means graphs of two variables.
At the end of the chapter, I'll take the notes copy from you.
So don't give me anything until it's over.
When is that going to be over?
We have four more sections to go.
So I guess right before spring break you give me
the notes for chapter 11.
All right, and then I'm thinking of making
copies of both chapters.
You get the-- I'm distributing them to you.
I haven't started and yet go ahead.
Could anybody tell me what the equation
that we used last time-- we proved it, actually.
What is the equation of the tangent plane
to a smooth surface or a patch of a surface at the point
m of coordinates x0, y0, z0, where the graph is
given by z equals f of x and y.
I'm going to label it on the patch of a surface.
OK, imagine it labeled brown there.
And can somebody tell me the equation of the other plane?
But because you have better memory,
being much younger, about 25 years younger than me or so.
So could you-- could anybody tell me what the tangent
planes equation-- I'll start.
And it's going to come to you.
z minus z0 equals.
And now let's see.
I'll pick a nice color.
I'll wait.
STUDENT: fx of x.
PROFESSOR TODA: f sub x, the partial derivative measured
at f0 i0 times the quantity x minus x0 plus--
STUDENT: f sub y.
PROFESSOR TODA: f sub y, excellent.
f sub y.
STUDENT: x0, y0.
PROFESSOR TODA: x0, y0 times y minus y0.
OK.
All right.
Now thinking of what those quantities mean, x minus x0, y
minus y0, z minus z0, what are they?
They are small displacements, aren't they?
I mean, what does it mean small displacement?
Imagine that you are near the point on both surfaces.
So what is a small neighborhood--
what's a typical small neighborhood [INAUDIBLE]?
It's a disk, right?
There are many kinds of neighborhoods, but one of them,
I'd say, would be this open disk, OK?
I'll draw that.
Now, if I have a red point-- I don't
know how to do that pink point-- somewhere nearby in planes--
this is the plane.
In plane, I have this point that is close.
And that point is xyz.
And you think, OK, can I visualize that better?
Well, guys, it's hard to visualize that better.
But I'll draw a triangle [? doing ?] a better job.
That's the frame.
This is a surface.
Imagine it's a surface, OK?
That's the point of x0, y0.
[? It's ?] the 0 and that.
Where is the point xyz again?
The point xyz is not on the pink stuff.
This is a pink surface.
It looks like Pepto Bismol or something.
You shaded it.
No.
That's not what I want.
I want the close enough point on the blue plane.
It's actually in the blue plane pie and this guy would be xyz.
So now say, OK, how far I x be from x0?
Well, I don't know.
We would have to check the points, the set 0,
check the blue point.
This is x.
So between x and x0, I have this difference,
which is delta x displacement, displacement along the x-axis,
away from the point, fixed point.
This is the fixed point, this point.
This point is p.
OK.
y minus y0, let's call that delta y, which
is the displacement along the y-axis.
And then the z minus z0 can be.
Just because I'm a mathematician and I don't like writing down
a lot, I would use s batch as I can,
compact symbols, to speed up my computation.
So I can rewrite this whole thing
as a delta z equals f sub x, x0 y0, which is a number.
It's a slope.
We discussed about that last time.
We even went skiing last time, when
we said that's like the slope in-- what's the x direction?
Slope in the x direction and slope in the y direction
on the graph that was the white covered with snow hill.
That was what we had last time.
Delta x plus f sub 0, another slope
in the y direction, delta y.
And fortunately-- OK, the book is a very good book, obviously,
right?
But I wish we could've done certain things better in terms
of comparisons between this notion in Calc III
and some corresponding notion in Calc I.
So you're probably thinking, what the heck
is this witch thinking about?
Well, I'm thinking of something that you
may want to remember from Calc I.
And that's going to come into place beautifully
right now because you have the Calc I, Calc III comparison.
And that's why it would be great-- the books don't even
talk about this comparison.
In Calc I, I reminded you about Mr. Leibniz.
He was a very nice guy.
I have no idea, right?
Never met him.
One of the fathers of calculus.
And he introduced the so-called Leibniz notation.
And one of you in office hours last Wednesday
told me, so the Leibnitz notation
for a function g of x-- I'm intentionally
changing notation-- is what?
Well, this is just the derivative
which is the limit of the different quotients
of your delta g over delta x-- as done by some
blutches-- 0, right, which would be the same as lim
of g of x minus g of x0 over x minus x0 as x approaches x0,
right?
Right.
So we've done that in Calc I. But it was a long time ago.
My mission is to teach you all Calc III,
but I feel that my mission is also
to teach you what you may not remember very well from Calc I,
because everything is related.
So what was the way we could have written this,
not delta g over delta x equals g prime.
No.
But it's an approximation of g prime around a very small
[INAUDIBLE], very close to x0.
So if you wanted to rewrite this approximation,
how would you have rewritten it?
Delta g--
STUDENT: g prime sub x.
PROFESSOR TODA: g prime of x0 times delta x.
OK?
Now, why this approximation?
What if I had put equal?
If I had put equal, it would be all nonsense.
Why?
Well, say, Magdalena, if you put equal, it's another object.
What object?
OK.
Let's look at the objects.
Let's draw a picture.
This is g.
This is x0.
This is g of x.
What's g prime?
g prime-- thank god-- is the slope of g prime x0 over here.
So if I want to write the line, the line is exactly this.
The red object is the line.
So what is the red object again?
It's y minus y over x minus x0 equals m, which
is g prime number 0.
m is the slope.
That's the point slope formula, thank you very much.
So the red object is this.
This is the line.
Attention is not the same.
The blue thing is my curve, more precisely
a tiny portion of my curve.
This neighborhood around the point is what I have here.
What I'm actually-- what?
I'm trying to approximate my curve
function with a little line.
And I say, I would rather approximate with a red line
because this is the best approximation
to the blue arc of a curve which is on the curve, right?
So this is what it is is just an approximation
of a curve, approximation of a curve of an arc of a curve.
But Magdalena's lazy today-- approximation
of an arc of a curve with a segment of a line,
with a segment of the tangent line
of the tangent [INAUDIBLE].
How do we call such a phenomenon?
An approximation of an arc of a circle
with a little segment of a tangent line
is like a discretization, right?
But we call it linear approximation.
It's called a linear approximation.
A-P-P, approx.
Have you ever seen a linear approximation
before coming from Calc II?
Well, in Calc II you've seen the Taylor's formula.
What is the Taylor's formula?
It's a beautiful thing that said what?
I don't know.
Let's remember together.
So relationship with Calc II, I'm
going to go and make an arrow-- relationship with Calc II,
because everything is actually related.
In Calc II-- how did we introduce Taylor's formula?
Well, instead of little a that you're so used to in Calc II,
we are going to put x0 is the same thing, right?
So what was Taylor's formula saying?
You have this kind of smooth, beautiful curve.
But being smooth is not enough.
You have that real analytic.
Real analytic means that the function can be
expanded in Taylor's formula.
So what does it mean?
It means that we have f of x prime is f of x0 equals-- or g.
You want-- it doesn't matter.
f prime of x0 times x minus x0 plus
dot, dot, dot, dot something that I'm going to put.
This is [? O. ?] It's a small quantity that's maybe not
so small, but I declare it to be negligible.
And so they're going to be negligible.
I have to make a face, a smiley face and eyes,
meaning that it's OK to neglect the second order
term, the third order term.
So what happens, that little h, when I square it,
say the heck with it.
It's going to be very small.
Like if h is 0.1 and then h squared will be 0.0001.
And I have a certain range of error that I allow,
a threshold.
I say that's negligible.
If h squared and h cubed and h to the fourth are negligible,
then I'm fine.
If I take all the other spot, that's
the linear approximation.
And that's exactly what I wrote here
with little g instead of f.
The only difference is this is little f and this is little g.
But it's the same exact formula, linear approximation.
Do you guys remember then next terms of the Taylor's formula?
STUDENT: fw--
PROFESSOR TODA: fw--
STUDENT: w over--
PROFESSOR TODA: So fw prime at x0 over--
STUDENT: 1 factorial.
PROFESSOR TODA: 2 factorial.
This was 1 factorial.
This was over 1 factorial.
But I don't write it because it's one.
STUDENT: Right.
PROFESSOR TODA: Here I would have f double prime of blah,
blah, blah over-- what did you say-- 2 factorial times x
minus x0 squared plus, plus, plus, the cubic [INAUDIBLE]
of the-- this is the quadratic term that I neglect, right?
So that was Taylor's formula.
Do I mention anything about it now?
We should.
But practically, the authors of the book
thought, well, everything is in the book.
You can go back and forth.
It's not like that unless somebody opens your eyes.
For example, I didn't see that when I was 21.
I couldn't make any connection between these Calc I,
Calc II, Calc III notions.
Because nobody told me, hey, Magdalena, open your eyes
and look at that in perspective and make
a comparison between what you learned in different chapters.
I had to grow.
After 20 years, I said, oh, I finally
see the picture of linearization of a function of, let's say,
n variables.
So all these total differentials will come in place
when time comes.
You have a so-called differential in Calc I.
And that's not delta g.
Some people say, OK, no, that's delta g.
No, no, no, no.
The delta x is a displacement.
The delta g is the induced displacement.
If you want this to be come a differential,
then you shrink that displacement
to infinitesimally small.
OK?
So it's like going from a molecule to an atom
to an electron to subatomic particles but even more,
something infinitesimally small.
So what do we do?
We shrink delta x into dx which is infinitesimally small.
It's like the notion of God but microscopically
or like microbiology compared to the universe, OK?
So dx is multiplied by g prime of x0.
And instead of delta g, I'm going to have a so-called dg,
and that's a form.
In mathematics, this is called a form or a one form.
And it's a special kind of object, OK?
So Mr. Leibniz was very smart.
He said, but I can rewrite this form like dg dx equals g prime.
So if you ever forget about this form which
is called differential, differential form,
you remember Mr. Leibniz, he taught you
how to write the derivative in two different ways, dg dx or g
prime.
What you do is just formally multiply g prime by dx
and you get dg.
Say it again, Magdalena-- multiply g prime by dx
and you get dg.
And that's your so-called differential.
Now, why do you say total differential-- total
differential, my god, like complete differentiation?
In 11.4, we deal with functions of two variables.
So can we say differentials?
Mmm, it's a little bit like a differential
with respect to what variable?
If you say with respect to all the variables,
then you have to be thinking to be smart and event,
create this new object.
If one would write Taylor's formula,
there is a Taylor's formula that we don't give.
OK.
Now, you guys are looking at me with excitement.
For one point extra credit, on the internet,
find Taylor's formula for n variables, functions
of n variables or at least two variables,
which was going to look like z minus z0 equals
f sub x at the point x0 at 0 times x minus x0 plus
f sub y at x0 y0 times x minus x0 plus second order terms
plus third order terms plus fourth order terms.
And the video cannot see me.
So what do we do?
We just truncate this part of Taylor's I say,
I already take the Taylor polynomial of degree one.
And the quadratic terms and everything else, the heck
with that.
And I call that a linear approximation,
but it's actually Taylor's formula being discussed.
We don't tell you in the book because we
don't want to scare you.
I think we would better tell you at some point,
so I decided to tell you now.
All right.
So this is Taylor's formula for functions of two variables.
We have to create not out of nothing
but out of this the total differential.
Who tells me?
Shrink the displacement, Magdalena.
The delta x shrunk to an infinitesimally small
will be dx.
Delta y will become dy.
The line is a smiley from the skies, just looking at us.
He loves our notations.
And this is dz.
So I'm going to write dz or df's the same thing equals f sub x.
At the point, you could be at any point
you are taking in particular, dx plus f sub y xy dy.
So this is at any point at the arbitrary point xy
in the domain where your function e is at least c1.
What does it mean, c1?
It means the function is differentiable
and the partial derivatives are continuous.
I said several times, I want even more than that.
I want it maybe second order derivatives
to exist and be continuous and so on and so forth.
And I will assume that the function can
be expanded [INAUDIBLE] series.
All right, now example of a final problem
that was my first problem on the final many times
and also on the common final departmental final.
And many students screwed up, and I
don't want you to ever make such a mistake.
So this is a mistake not to make, OK, mistake not
to make because after 20 something years of teaching,
I'm quite familiar with the mistakes students
make in general and I don't want you to make them.
You are too good to do this.
So problem 1.
On the final, I said-- we said-- the only difference was
on some departmental finals, we gave a more sophisticated
function.
I'm going to give only some simple function
for this polynomial.
That's beautiful.
And then I said we said write the differential
of this function at an arbitrary point x, y.
And done.
And [INAUDIBLE].
Well, let me tell you what some of my students-- some
of my studentss-- don't do that.
I'm going to cross it with red.
And some of my students wrote me very beautifully df
equals 2x plus 2y.
And that can send me to the hospital.
If you want to go to the ER soon, do this on the exam
because this is nonsense.
Why is this nonsense?
This is not--
STUDENT: [INAUDIBLE] dx or dy.
PROFESSOR TODA: Exactly.
So the most important thing is that the df is like-- OK,
let me come back to driving.
I'm driving to Amarillo-- and I give this example to my calc 1
students all the time because it's a linear motion in terms
of time.
And let's say I'm on cruise control or not.
It doesn't matter.
When we drive and I'm looking at the speedometer and I see 60--
I didn't want to say more, but let's say 80, 80 miles an hour.
That is a miles an hour.
That means the hour is a huge chunk delta h or delta t.
Let's call it delta t because it's time.
I'm silly.
Delta t is 1.
Delta s, the space, the space, is going
to be the chunk of 60 miles.
But then that is the average speed that I had.
So that's why I said 60.
That's the average speed I had in my trip,
during my trip [INAUDIBLE].
There were moments when my speed was 0 or close to 0.
Let's assume it was never 0.
But that means there were many moments when my speed could've
been 100, and nobody knows because they didn't catch me.
So I was just lucky.
So in average, if somebody is asking you what is the average,
that doesn't tell them anything.
That reminds me of that joke-- overall I'm good,
the statistician joke who was, are you cold?
Are you warm?
And he was actually sitting on with one half of him
on a block of ice and the other half on the stove,
and he says, in average, I'm fine.
But he was dying.
This is the same kind of thing.
My average was 60 miles an hour, but I almost
got caught when I was driving almost 100.
But nobody knows because I'm not giving you that information.
That's the infinitesimally small information that I have not
put correctly here means that what is
what I see on the speedometer?
It's the instantaneous rate of change
that I see that fraction of second.
So that means maybe a few feet per a fraction of a second.
It means how many feet did I travel
in that fraction of a second?
And if that fraction of a second is very tiny that I cannot even
express it properly, that's what I'm going to have--
df equals f prime dx.
So df and dx have to be small because their ratio will be
a good number, like 60, like 80, but [? them in ?] themselves
delta m delta [? srv, ?] very tiny things.
It's the ratio that matters in the end to be 60, or 80,
or whatever.
So I have 2x dx plus 2y dy.
Never say that the differential, which
is something infinitesimally small,
is equal to this scalar function that it doesn't even
make any sense.
Don't do that because you get 0 points
and then we argue, and I don't want
you to get 0 points on this problem, right.
So it's a very simple problem.
All I want to test you on would be this definition.
Remember, you're going to see that again on the midterm
and on the final, or just on the final.
Any questions about that?
All right.
So I want to give you the following homework out
of section 11.4 on top of the web work.
Read all the solved examples of the section.
OK.
So for example, somebody tells you
I have to apply this knowing that I have
an error of measurement of some sort in the s direction
and an error of measurement of some sort in the y direction.
There are two or three examples like that.
They will give you all this data, including the error
measurement.
For delta, it should be 0.1.
Don't confuse the 0.1 with dx. dx is not a quantity.
dx is something like micro cosmic thing.
It's like infinitely [? small ?].
Infinitesimally small.
So saying that dx should be 0.1 doesn't make any sense,
but delta x being 0.1 make sense.
Delta y being 0.3 makes sense.
And they ask you to plug it in and find
the general difference.
For example, where could that happen?
And you see examples in the book.
Somebody measures something-- an area of a rectangle
or a volume of a cube.
But when you measure, you make mistakes.
You have measurement errors.
In the delta x, you have an error of plus minus 0.1.
In the y direction, you have displacement error 0.2 or 0.3,
something like that.
What is the overall error you are
going to make when you measure that function of two variables?
That's what you have.
So you plug in all those displacements
and you come up with the computational problem.
Several of you Wednesday we discussed in my office already
solved those problems through web work and came to me,
and I said, how did you know to plug in those [? numbers ?]?
Well, it's not so hard.
It's sort of common sense.
Plus, I looked in the book and that gave me the idea
to remind you to look in the book
for those numerical examples.
You will have to use your calculator.
So you don't have it with you, you generally, we
don't use in the classroom, but it's very easy.
All you have to do is use the calculator and [INAUDIBLE]
examples and see how it goes.
I wanted to show you something more interesting
even, more beautiful regarding something
we don't show in the book until later on,
and I'm uncomfortable with the idea of not showing this to you
now.
An alternate way, or more advanced way,
more advanced way, to define the tangent plane--
the tangent plane-- to a surface S at the point p.
And I'll draw again.
Half of my job is drawing in this class, which I like.
I mean, I was having an argument with one of my colleagues who
said, I hate when they are giving me to teach calculus 3
because I cannot draw.
I think that the most beautiful part
is that we can represent things visually,
and this is just pi, the tangent plane I'm after,
and p will be a coordinate 0 by 0, z0.
And what was the label?
Oh, the label.
The label.
The label was internal where z equals f of xy.
But more generally, I'll say this time plus more generally,
what if you have f of xyz equals c for that surface.
Let's call it [INAUDIBLE].
F of xy is [INAUDIBLE].
And somebody even said, can you have a parametrization?
And this is where I wanted to go.
Ryan was the first one who asked me,
but then there were three more of you
who have restless minds plus you--
because that's the essence of being active here.
We don't lose our connections.
We lose neurons anyway, but we don't lose our connections
if we think, and anticipate things,
and try to relate concepts.
So if you don't want to get Alzheimer's, just
think about the parametrization.
So can I have a parametrization for a surface?
All righty, what do you mean?
What if somebody says for a curve, we have r of t, right,
which was what?
It was x of ti plus y of tj plus z of tk, and we were so happy
and we were happy because we were traveling
in time with respect to the origin,
and this was r of t at time t.
[INAUDIBLE]
But somebody asked me, [INAUDIBLE],
can you have such a position vector moving on a surface?
Like look, it's a rigid motion.
If you went to the robotics science
fair, Texas Tech, or something like that, you know about that.
Yeah, cities.
So how do we introduce such a parametrization?
We have an origin of course.
An origin is always important.
Everybody has an origin.
And I take that position vector, and where does it start?
It starts at the origin, and the tip of it is on the surface,
And it's gliding on the surface, the tip of it.
And that's going to be r, but it's not going to be r of t.
It's going to be r of longitude and latitude.
Like imagine, that would be the radius coming
from the center of the earth.
And it depends on two parameters.
One of them would be latitude.
Am I drawing this right?
Latitude--
STUDENT: [INAUDIBLE] longitude.
PROFESSOR TODA: --from a latitude 0.
I'm at the equator.
Then latitude 90 degrees.
I'm at the North Pole.
In mathematics, we are funny.
We say latitude 0, latitude 90 North Pole,
latitude negative 90, which is South Pole.
And longitude from 0 to 2 pi.
Meridian 0 to all around.
So r will be not a function of t but a function of u and b,
thank god, because u and b are the latitude and longitude
sort of.
So we have x of uv i plus y of uv j plus z of uv k.
You can do that.
And you say, but can you give us an example, because this
looks so abstract for god sake.
If you give me the graph the way you gave it to me
before z equals f of xy, please parametrize this for me.
Parametrize it for me because I'm lost.
You are not lost.
We can do this together.
Now what's the simplest way to parametrize
a graph of the type z equals f of xy?
Take the xy to be u and v. Take x
and y to be your independent variables
and take z to be the dependent variable.
I'm again expressing these things in terms of variables
like I did last time.
Then I say, let's take this kind of parametrization. [INAUDIBLE]
vu, right.
y would be v. Then I'm going to write r of x and y
just like that guy will be [INAUDIBLE] of xn.
[? y ?] will say, wait a minute.
I will have to re-denote everybody with capitals.
Then my life will become better because you
don't have to erase.
You just make little x big, little y bigs,
bigs, big, capitalized XYZ.
And then I'll say OK, XYZ will be my setting here in 3D.
All right.
So how am I going to re-parametrize
the whole surface?
Whole surface will be r of xy equals in this case, well,
let's think about it.
In this case, I'm going to have xy.
And where's the little f?
I just erased it.
I was smart, right, that I erased f of xy.
So I have x, y, and z, which is f of xy.
And this is the generic point p of coordinates xy f of xy.
So I say, OK, what does it mean?
I will project this point.
And this is the point when big x becomes little
x, when big y becomes-- where is my y-axis?
Somebody ate my y axis.
[INAUDIBLE]
So when big Y becomes little y, little y
is just an instance of big Y. And big Z will take what value?
Well, I need to project that.
How do you project from a point to the z-axis?
You have to take the parallel from the point
to the horizontal plane until you
hit the-- [INAUDIBLE] the whole plane parallel to the floor
through the point p.
And what do I get here?
STUDENT: [INAUDIBLE].
PROFESSOR TODA: Not z0, but it's little z
equals f of xy, which is an instance of the variable xz.
For you programmers, you know that big z will be a variable
and little z will be [INAUDIBLE] a variable.
OK.
So I parametrized my graph in a more general way,
general parametrization for a graph.
And now, what are-- what's the meaning of r sub x and r sub y?
What are they?
STUDENT: [INAUDIBLE].
PROFESSOR TODA: Now, we don't say that in the book.
Shame on us.
We should have because I was browsing through the projects
about a year and a half ago.
The senior projects of a few of my students
who are-- two of them were in mechanical engineering.
One of them was in petroleum engineering.
And he actually showed me that they were doing this.
They were taking vectors that depend on parameters--
this is a vector [INAUDIBLE]-- and differentiated them with
respect to those parameters.
And I was thinking OK, did we do the partial derivatives r sub
x, r sub y?
Not so much.
But now I want to do it because I think that prepares
you better as engineers.
So what is r sub x and what is r sub y?
And you say, well, OK. [INAUDIBLE],
I think I know how to do that in my sleep, right.
If you want me to do that theoretically
from this formula, but on the picture,
I really don't know what it is.
So I'm asking you what I'm going to have in terms
of r sub x and r sub y.
They will be vectors.
This should be a vector as well, right.
And for me, vector triple means the identification
between the three coordinates and the physical vector.
So this is the physical vector.
Go ahead and write x prime with respect to x is 1.
y prime with respect to x is 0.
The third [INAUDIBLE] prime with respect
to x is just whatever this little f is,
it's not any of my business.
It's a [INAUDIBLE] function f sub x.
Well, what is the second vector?
STUDENT: 0, 1, f sub y.
PROFESSOR TODA: 0, 1, f sub y.
Now, are they slopes?
No.
These are slopes.
That's a slope and that's a slope.
And we learned about those in 11.3,
and we understood that those are ski slopes, they were.
In the direction of x and the direction of y,
the slopes of the tangents to the coordinate lines.
But this looks like I have a direction of a line,
and this would be the lope, and that's the direction of a line,
and that would be the slope.
What are those lines?
STUDENT: [INAUDIBLE] to the function [INAUDIBLE].
PROFESSOR TODA: Let me draw.
Then shall I erase the whole thing?
No.
I'm just going to keep-- I'll erase the tangent.
Don't erase anything on your notebooks.
So this is the point p.
It's still there.
This is the surface.
It's still there.
So my surface will be x, slices of x, [? S ?] constant
are coming towards you.
They are these [? walls ?] like that, like this, yes.
It's like the CT scan.
I think that when they slice up your body,
tch tch tch tch tch tch, take pictures
of the slices of your body, that's the same kind of thing.
So x0, x0, x0, x0.
I'm going to [INAUDIBLE] planes and I had x equals x0.
And in the other direction, I cut and I get, what do I get?
Well, I started bad.
Great, Magdalena, this is--
What is this pink?
It's not Valentine's Day anymore. y equals [INAUDIBLE].
And this is the point.
So, as Alex was trying to tell you,
our sub x would represent the vector, the physical vector
in 3D, that is originating at p and tangent to which
of the two, to the purple one or to the red one?
STUDENT: Red.
Uh, purple.
PROFESSOR TODA: Make up your mind.
STUDENT: The purple one.
PROFESSOR TODA: [INAUDIBLE] constant and [INAUDIBLE]
constant in the red one, y equals y0, right?
So, this depends on x.
So this has r sub x.
This is the velocity with respect to the variable x.
And the other one, the blue one, x equals x0,
means x0 is held fixed and y is the variable.
So I have to do r sub y, and what am I gonna get?
I'm gonna get the blue vector.
What's the property of the blue vector?
It's tangent to the purple line.
So r sub y has to be tangent to the curve.
x0, y, f of x0 and y is the curve.
And r sub x is tangent to which curve?
Who is telling me which curve?
x, y0 sub constant, f of x and y0.
So that's a curve that depends only on y,
y is the time in this case.
And that's the curve that depends only on x.
x is the time in this case.
r sub x and r sub y are the tangent vectors.
What's magical about them?
If I shape this triangle between them,
that will be the tangent plane.
And I make a smile because I discovered the tangent plane
in a different way than we did it last time.
So the tangent plane represents the plane of the vector r sub
x and r sub y.
The tangent plane represents the plane
given by vectors r sub x and r sub y with what conditions?
It's a conditional.
r sub x and r sub y shouldn't be 0.
r sub x different from 0, r sub y different from 0,
and r sub x and r sub y are not collinear.
What's gonna happen if they are collinear?
Well, they're gonna collapse; they are not
gonna determine a plane.
So there will be no tangent planes.
So they have to be linearly independent.
For the people who are taking now linear algebra, I'm saying.
So we have no other choice, we have
to assume that these vectors, called partial velocities,
by the way, for the motion across the surface.
OK?
These are the partial velocities, or partial velocity
vectors.
Partial velocity vectors have to determine a plane,
so I have to assume that they are non-zero,
they never become 0, and they are not collinear.
If they are collinear, life is over for you.
OK?
So I have to assume that I throw away all the points where
the velocities become 0, and all the points where--those are
singularity points--where my velocity vectors are 0.
Have you ever studied design?
Any kind of experimental design.
Like, how do you design a car, the coordinate lines on a car?
I'm just dreaming.
You have a car, a beautiful car, and then you have-- Well,
I cannot draw really well, but anyway.
I have these coordinate lines on this car.
It's a mesh what I have there.
Actually, we do that in animation all the time.
We have meshes over the models we have in animation.
Think Avatar.
Now, those are all coordinate lines.
Those coordinate lines would be, even your singularities, where?
For example, if you take a body in a mesh like that, in a net,
in, like, a fishnet, then you pull from the fishnet,
all the coordinate lines will come together,
and this would be a singularity.
We avoid this kind of singularity.
So these are points where something bad happened.
Either the velocity vectors become collinear.
You see what I'm talking about?
Or the velocity vectors shrank to 0.
So that's a bad point; that's a singularity point.
They have this problem when meshing.
So when they make these models that
involve two-dimensional meshing and three-dimensional ambient
space, like it is in animation, the mesh
is called regular if we don't have
this kind of singularity, where the velocity vectors become 0,
or collinear.
It's very important for a person who programs in animation
to know mathematics.
If they don't understand these things, it's over.
Because you write the matrix, and you will know the vectors
will become collinear when the two vectors--let's say two rows
of a matrix--
STUDENT: Parallel.
PROFESSOR TODA: Are proportional.
Or parallel.
Or proportional.
So, everything is numerical in terms of those matrices,
but it's just a discretization of a continuous phenomenon,
which is this one.
Do you remember Toy Story?
OK.
The Toy Story people, the renderers,
the ones who did the rendering techniques for Toy Story,
both have their master's in mathematics.
And you realize why now to do that you
have to know calc I, calc II, calc III, linear algebra,
be able to deal with matrices.
Have a programming course or two; that's essential.
They took advanced calculus because some people
don't cover thi-- I was about to skip it right now in calc III.
But they teach that in advanced calculus 4350, 4351.
So that's about as far as you can get,
and differential equation's also very important.
So, if you master those and you go into something else,
like programming, electrical engineering,
you're ready for animation.
[INAUDIBLE] If you went I want to be a rendering
guy for the next movie, then they'll say no,
we won't take you.
I have a friend who works for Disney.
She wanted to get a PhD.
At some point, she changed her mind
and ended up just with a master's in mathematics
while I was in Kansas, University of Kansas,
and she said, "You know what?
Disney's just giving me $65,000 as an intern."
And I was like OK and probably asked [INAUDIBLE] $40,000 as
a postdoc.
And she said, "Good luck to you."
Good luck to you, too.
But we stayed in touch, and right now she's
making twice as much as I'm making, for Disney.
Is she happy?
Yeah.
Would I be happy?
No.
Because she works for 11 hours a day.
11 hours a day, on a chair.
That would kill me.
I mean, I spend about six hours sitting on a chair every day
of the week, but it's still too much.
She's a hard worker, though.
She loves what she's doing.
The problem is your eyes.
After a while, your eyes are going bad.
So, what is the normal for the plane in this case?
I'll try my best ability to draw normal.
The normal has to be perpendicular
to the tangent space, right?
Tangent plane.
So, n has to be perpendicular to our sub
x and has to be perpendicular to our sub y.
So, can you have any guess how in the world
I'm gonna get n vector?
STUDENT: [INAUDIBLE]
PROFESSOR TODA: [INAUDIBLE] That's
why you need to know linear algebra
sort of at the same time, but you guys are making it fine.
It's not a big deal.
You have a matrix, i, j, k in the front row vectors,
and then you have r sub x that you gave me, and I erased it.
1, 0, f sub x.
0, 1, f sub y.
And you have exactly 18 seconds to compute this vector.
STUDENT: [INAUDIBLE]
PROFESSOR TODA: You want k, but I want to leave k at the end
because I always order my vectors.
Something i plus something j plus something k.
[INTERPOSING VOICES]
PROFESSOR TODA: Am I right?
Minus f sub x--
STUDENT: Minus f of x plus k.
PROFESSOR TODA: --times i.
For j, do I have to change sign?
Yeah, because 1 plus 2 is odd.
So I go minus 1.
And do it slowly.
You're not gonna make fun of me; I gotta make fun of you, OK?
And minus 1 times--
STUDENT: Did you forget f y?
PROFESSOR TODA: --f sub y--I go like that--sub y times j plus
k.
As you said very well in the most elegant way
without being like yours, but I say it like this.
So you have minus f sub x, minus f sub y,
and 1 as a triple with angular brackets--You love that.
I don't; I like it parentheses [INAUDIBLE]--equals n.
But n is non-unitary, but I don't care.
Why don't I care?
I can write the tangent plane very well
without that n being unitary, right?
It doesn't matter in the end.
These would be my a, b, c.
Now I know my ABC.
I know my ABC.
So, the tangent plane is your next guess.
The tangent plane would be perpendicular to n.
So this is n.
The tangent plane passes through the point p
and is perpendicular to n.
So, what is the equation of the tangent plane?
STUDENT: Do you want scalar equations?
PROFESSOR TODA: A by x minus 0.
Very good.
That's exactly what I wanted you to write.
All right, so, does it look familiar?
Not yet.
[STUDENT SNEEZES]
STUDENT: Bless you.
PROFESSOR TODA: Bless you.
Who sneezed?
OK.
Am I almost done?
Well, I am almost done.
I have to go backwards, and whatever I get
I'll put it big here in a big formula on top.
I'm gonna say oh, my God.
No, that's not what I'm gonna say.
I'm gonna say minus f sub x at my point p--that is a, right?
Times x minus x0.
Plus minus f sub y at the point p; that's b.
y minus y0 plus--c is 1, right?
c is 1.
I'm not gonna write it because if I write
it you'll want to make fun of me. z minus z0 equals 0.
Now it starts looking like something familiar, finally.
Now we discovered that the tangent plane
can be written as z minus z0.
I'm keeping the guys z minus z0 on the left-hand side.
And these guys are gonna move to the right-hand side.
So, I'm gonna have again, my friend,
the equation of the tangent plane for the graph z equals f
of x,y.
But you will say OK, I think by now
we've learned these by heart, we know
the equation of the tangent plane, and now we're asleep.
But what if your surface would be implicit the way
you gave it to us at first.
Maybe you remember the sphere that was an implicit equation,
x squared plus x squared plus x squared equals--
What do you want it to be?
STUDENT: 16.
PROFESSOR TODA: Huh?
STUDENT: 16.
PROFESSOR TODA: 16.
So, radius should be 4.
And in such a case, the equation is of the type f of x, y, z
equals constant.
Can we write again the equation [INAUDIBLE]?
Well, you say well, you just taught
us some theory that says I have to think of u and v, but not x
and y.
Because if I think of x and y, what would they be?
I think the sphere as being an apple.
Not an apple, something you can cut easily.
Well, an apple, an orange, something.
A round piece of soft cheese.
I started being hungry, and I'm dreaming.
So, this is a huge something you're gonna slice up.
If you are gonna do it with x and y,
the slices would be like this.
Like that and like this, right?
And in that case, your coordinate curves
are sort of weird.
If you want to do it in different coordinates,
so we want to change coordinates,
and those coordinates should be plotted to the longitude,
then we cannot use x and y.
Am I right?
We cannot use x and y.
So those u and v will be different coordinates,
and then we can do it like that, latitude.
[INAUDIBLE] minus [INAUDIBLE].
And longitude.
We are gonna talk about spherical coordinates
later, not today.
Latitude and longitude.
1 point extra credit, because eventually we
are gonna get there, chapter 12.7.
12.7 comes way after spring break.
But before we get there, who is in mechanical engineering
again?
You know about Euler's angles, and stuff like that.
OK.
Can you write me the equations of x
and y and z of the sphere with respect to u and v,
u being latitude and v being longitude?
These have to be trigonometric functions.
In terms of u and v, when u is latitude and v is longitude.
1 point extra credit until a week from today.
How about that?
U and v are latitude and longitude.
And express the xyz point in the ambient space on the sphere.
x squared plus x squared plus x squared would be 16.
So you'll have lots of cosines and sines [INAUDIBLE]
of those angles, the latitude angle and the longitude angle.
And I would suggest to you that you take--for the extra credit
thing--you take the longitude angle to be from 0 to 2pi,
from the Greenwich 0 meridian going back to himself,
and--well, there are two ways we do this in mathematics
because mathematicians are so diverse.
Some of us, say, for me, I measure the latitude
starting from the North Pole.
I think that's because we all believe in Santa or something.
So, we start measuring always from the North Pole
because that's the most important place on Earth.
They go 0, pi over 2, and then-- what is our lat--shame on me.
STUDENT: It's 33.
PROFESSOR TODA: 33?
OK.
Then pi would be the equator, and then pi
would be the South Pole.
But some other mathematicians, especially biologists
and differential geometry people, I'm one of them,
we go like that.
Minus pi over 2, South Pole 0, pi over 2 North Pole.
So we shift that kind of interval.
Then for us, the trigonometric functions of these angles
would be a little bit different when we
do the spherical coordinates.
OK, that's just extra credit.
It has nothing to do with what I'm gonna do right now.
What I'm gonna do right now is to pick a point on Earth.
We have to find Lubbock.
STUDENT: It's on the left.
PROFESSOR TODA: Here?
Is that a good point?
This is LBB.
That's Lubbock International Airport.
So, for Lubbock--let's call it p as well--draw the r sub u,
r sub v. So, u was latitude.
So if I fix the latitude, that means I fix
the 33 point whatever you said.
u equals u0.
It is fixed, so I have u fixed, and v equals v0 is that.
I fixed the meridian where we are.
What is this tangent vector?
To the pink parallel, the tangent vector
would be r sub what?
STUDENT: v.
PROFESSOR TODA: r sub v. You are right.
You've got the idea.
And the blue vector would be the partial velocity.
That's the tangent vector to the blue meridian,
which is r sub u.
And what is n gonna be? n's gonna be r sub u [INAUDIBLE].
But is there any other way to do it in a simpler way
without you guys going oh, man.
Suppose some of you don't wanna do the extra credit
and then say the heck with it; I don't
care about her stinking extra credit until chapter 12,
when I have to study the spherical coordinates,
and is there another way to get n.
I told you another way to get n.
Well, we are getting there.
n was the gradient of f over the length of that.
And if we want it unitary, the length of f was what?
f sub x, f sub y, f sub z vector, where
the implicit equation of the surface was f of x, y, z
equals c.
So now we've done this before.
You say Magdalena, you're repeating yourself.
I know I'm repeating myself, but I want you to learn this twice
so you can remember it.
What is f of x, y, z?
In my case, it's x squared plus y squared plus z squared
minus 16, or even nothing.
Because the constant doesn't matter anyway
when I do the gradient.
You guys are doing homework.
You saw how the gradient goes.
So gradient of f would be 2x times-- and that's
the partial derivative times i plus 2y times j plus 2z times
k-- that's very important.
[? Lovett ?] has some coordinates we plug in.
Now, can we write-- two things.
I want two things from you.
Write me a total differential b tangent plane
at the point-- so, a, write the total differential.
I'm not going to ask you you to do a linear approximation.
I could.
B, write the tangent plane to the sphere at the point
that-- I don't know.
I don't want one that's trivial.
Let's take this 0, square root of 8, and square root of 8.
I just have to make sure that I don't
come with some nonsensical point that's
not going to be on the sphere.
This will be because I plugged it in in my mind.
I get 8 plus 8 is 16 last time I checked, right?
So after we do this we take a break.
Suppose that this is a problem on your midterm,
or on your final or on your homework,
or on somebody [? YouTubed it ?] for a lot of money,
you asked them, $25 an hour for me to work that problem.
That's good.
I mean-- it's-- it's a class that you're taking
for your general requirement because your school wants you
to take calc 3.
But it gives you-- and I know from experience,
some of my students came back to me and said,
after I took calc 3, I understood it
so well that I was able to tutor calc 1, calc 2, calc 3,
so I got a double job.
Several hours a week, the tutoring center,
math department, and several hours
at the [INAUDIBLE] center.
You know what I'm talking about?
So I've had students who did well and ended up liking this,
and said I can tutor this in my sleep.
So-- and also private tutoring is always a possibility.
OK.
Write total differential.
df equals, and now I'll say at any point.
So I don't care what the value will be.
I didn't say at what point.
It means in general.
Why is that?
You tell me, you know that by now.
2x times what?
Now, you learned your lesson, you're
never gonna make mistakes.
2y plus 2z dz.
That is very good.
That's the total differential.
Now, what is the equation of the tangent plane?
It's not gonna be that.
Because I'm not considering a graph.
I'm considering an implicitly given surface
by this implicit equation f of x, y, z, equals c, your friend.
So what was, in that case, the equation of the plane
written as?
STUDENT: [INAUDIBLE]
PROFESSOR TODA: I'm-- yeah, you guys are smart.
I mean, you are fast.
Let's do it in general.
F sub x-- we did that last time, [INAUDIBLE] times--
do you guys remember?
x minus x0.
And this is at the point plus big F sub y at the point times
y minus y0 plus big F sub z at the point z minus z0.
This is just review.
Equals 0.
Stop.
Where do these guys come from?
From the gradient.
Which are the a,b,c, now I know my ABCs, from the normal.
My ABCs from the normal.
So in this case-- I don't want to erase
this beautiful picture.
The last thing I have to do before the break is-- you
said 0.
I'm a lazy person by definition.
Can you tell me why you said 0 times?
STUDENT: Because the x value is [INAUDIBLE]
PROFESSOR TODA: You said 2x, plug in and x equals 0
from your point, Magdalena, so you don't
have to write down everything.
But I'm gonna write down 0 times x minus 0 plus-- what's
next for me?
STUDENT: 2 square root 8.
PROFESSOR TODA: 2y, 2 root 8.
Is root 8 beautiful?
It looks like heck.
At the end I'm gonna brush it up a little bit.
This is the partial-- f sub y of t times y minus-- who is y, z?
Root 8.
Do I like it?
I hate it, but it doesn't matter.
Because I'm gonna simplify.
Plus again, 2 root 8, thank you.
This is my c guy.
Times z minus root 8 equals 0.
I picked another example from the one from the book,
because you are gonna read the book anyway.
I'm gonna erase that.
And I'm gonna brush this up because it
looks horrible to me.
Thank God this goes away.
So the plane will simply be a combination
of my y and z in a constant.
And if I want to make my life easier,
I'm gonna divide by what?
By this.
So in the end, it doesn't matter.
Come on.
I'll get y minus root 8 plus c minus root 8 equals 0.
Do I like it?
I hate it.
No, you know, I don't like it.
Why don't I like it?
It's not simplified.
So in any case, if this were multiple choice,
it would not be written like that, right?
So what would be the simplified claim in this case?
The way I would write it-- a y plus a z minus--
think, what is root 8?
STUDENT: 2 root 2.
PROFESSOR TODA: And 2 root 2.
And 2 root 2, how much-- minus 4 root 2.
And this is how you are expected to leave this answer boxed.
This is that tangent plane at the point.
To the sphere.
There are programs-- one time I was teaching
advance geometry, 4331, and one thing I gave my students to do,
which was a lot of fun-- using a parametrization,
plot the entire sphere with MathLab.
We did it with MathLab.
Some people said they know [INAUDIBLE] I didn't care.
So MathLab for me was easier, so we
plotted the sphere in MathLab.
We picked a point, and we drew-- well,
we drew-- with MathLab we drew the tangent plane that
was tangent to the sphere at that point.
And they liked it.
It was-- you know what this class is,
is-- if you're math majors you take it.
It's called advanced geometries.
Mainly it's theoretical.
It teaches you Euclidian axioms and stuff,
and then some non-Euclidian geometries.
But I thought that I would do it into an honors class.
And I put one third of that last class visualization
with MathLab of geometry.
And I think that was what they liked the most, not so
much the axiomatic part and the proofs,
but the hands-on computation and visualization in the lab.
We have this lab, 113.
We used to have two labs, but now we are poor,
we only have one.
No, we lost the lab.
The undergraduate lab-- 009, next to you,
is lost because-- I used to each calc 3 there.
Not because-- that's not why we lost it.
We lost it because we-- we put some 20 graduate students
there.
We have no space.
And we have 130 graduate students in mathematics.
Where do you put them?
We just cram them into cubicles.
So they made 20 cubicles here, and they put some,
so we lost the lab.
It's sad.
All right.
So that's it for now.
We are gonna take a short break, and we
will continue for one more hour, which is mostly application.
I'm sort of done with 11.4.
I'll jump into 11.5 next.
Take a short break.
Thanks for the attendance.
Oh, and you did the calculus.
Very good.
Did this homework give you a lot of headaches, troubles
or anything, or not?
Not too much?
It's a long homework.
49 problems-- 42 problems.
It wasn't bad?
OK, questions from the-- what was it, the first part-- mainly
the first part of chapter 11.
This is where we are.
Right now we hit the half point because 11.8
is the last section.
And we will do that, that's Lagrange multipliers.
So, let's do a little bit of a review.
Questions about homework.
Do you have them?
Imagine this would be office hour.
What would you ask?
STUDENT: I know it's a stupid question,
but my visualization [INAUDIBLE] coming along, and question
three about the sphere passing the plane and passing the line.
So you have a 3, 5, and 4 x, y, and z,
and you have a radius of 5.
Is it passing the x, y plane?
Is it passing [INAUDIBLE] x plane and [INAUDIBLE]
passing the other plane.
PROFESSOR TODA: So-- say again.
So you have 3 and 4 and 5--
STUDENT: x minus-- yes.
PROFESSOR TODA: What are the coordinates?
STUDENT: 3, 4, and 5.
PROFESSOR TODA: 3, 4, and 5, just as you said them.
You can--
STUDENT: And the radius is 5.
PROFESSOR TODA: Radius of?
STUDENT: 5.
Radius is equal to 5.
[INAUDIBLE]
PROFESSOR TODA: Yeah, well, OK.
So assume you have a sphere of radius 5, which
means you have 25.
If you do the 3 squared plus 4 squared plus 5 squared,
what is that?
For this point.
You have two separate points.
For this point you have 25 plus 25.
Are you guys with me?
So you have the specific x0, y0, z0.
You do the sum of the squares, and you get 50.
My question is, is this point outside, inside the sphere
or on the sphere?
On the sphere, obviously, it's not,
because it does not verify the equation of the sphere, right?
STUDENT: [INAUDIBLE] those the location of the center point.
STUDENT: Where's the center of the sphere?
STUDENT: [INAUDIBLE]
PROFESSOR TODA: The center of the sphere would be at 0.
STUDENT: [INAUDIBLE]
PROFESSOR TODA: We are making up a question.
So, right?
So practically, I am making up a question.
STUDENT: Oh, OK.
PROFESSOR TODA: So I'm saying if you have a sphere of radius 5,
and somebody gives you this point of coordinates 3, 4,
and 5, where is the point?
Is it inside the sphere, outside the sphere or on the sphere?
On the sphere it cannot be because it doesn't verify
the sphere.
Ah, it looks like a Mr. Egg.
I don't like it.
I'm sorry, it's a sphere.
So a point on a sphere that will have-- that's a hint.
A point on a sphere that will have coordinates 3 and 4
would be exactly 3, 4, and 0.
So it would be where?
STUDENT: 16, 4.
PROFESSOR TODA: 3 squared plus 4 squared is 5 squared, right?
So those are Pythagorean numbers.
That's the beauty of them.
I'm trying to draw well.
Right.
This is the point a.
You go up how many?
You shift by 5.
So are you inside or outside?
STUDENT: Outside.
PROFESSOR TODA: Yeah.
STUDENT: Are you outside or are you exactly on-- oh.
Sorry, I thought--
PROFESSOR TODA: You go--
STUDENT: I thought you were saying point a.
Point a is like exactly-- [INAUDIBLE]
PROFESSOR TODA: You are on the equator,
and from the Equator of the Earth,
you're going parallel to the z-axis, then you stay outside.
But the question is more subtle than that.
This is pretty-- you figured it out.
1 point-- 0.5 extra credit.
That we don't have-- I wish we had-- maybe
we'll find some time.
When I-- when we rewrite the book, maybe we should do that.
So express the points outside the sphere, inside the sphere,
and on the sphere using exclusively
equalities and inequalities.
And that's extra credit.
So, of course, the [INAUDIBLE] is obvious.
The sphere is the set of the triples x, y, z in R3.
OK, I'm teaching you a little bit of mathematical language.
x, y, z belongs to R3, R3 being the free space,
with the property that x squared plus y squared plus z squared
equals given a squared.
What if you have less than, what if you have greater than?
Ah, shut up, Magdalena.
This is all up to you.
You will figure out how the points
on the outside and the points on the inside are characterized.
And unfortunately we don't emphasize that in the textbook.
I'll erase.
You figured it out.
And now I want to move on to something
a little bit challenging, but not very challenging.
STUDENT: Professor, [INAUDIBLE]
PROFESSOR TODA: The last requirement
on the extra credit?
So I said the sphere represents the set of all
triples x, y, z in R3 with the property
that x squared plus y squared plus y squared plus z squared
equals a squared.
With the equality sign.
Represent the points on the inside of the sphere
and the outside of the sphere using just inequalities.
Mathematics.
No writing, no words, just mathematics.
In set theory symbols.
Like, the set of points with braces like that.
OK.
I'll help you review a little bit of stuff from the chain
rule in-- in chapter-- I don't know, guys,
it was a long time ago.
Shame on me.
Chapter 3, calc 1.
Versus chain rule rules in calc in-- chapter 5 calc 3.
This is a little bit of a warmup.
I don't want to [INAUDIBLE] again
next time when we meet on Thursday.
Bless you.
The bless you was out of the context.
What was the chain rule?
We did compositions of functions,
and we had a diagram that we don't show you, but we should.
There is practically a function that comes from a set A
to a set B to a set C. These are the sets.
And we have g and an f.
And we have g of f of t.
t is your favorite letter here.
How do you do the derivative with respect
to g composed with f?
I asked the same question to my Calc 1 and Calc 2 students,
and they really had a hard time expressing themselves,
expressing the chain rule.
And when I gave them an example, they
said, oh, I know how to do it on the example.
I just don't know how to do it on the-- I like the numbers,
but I don't like them letters.
So how do we do it in an example?
I chose natural log, which you find everywhere.
So how do you do d dt of this animal?
It's an animal.
STUDENT: [INAUDIBLE]
PROFESSOR TODA: So the idea is you go from the outside
to the inside, one at a time.
My students know that.
You prime the function, the outer function,
the last one you applied, to the function inside.
And you prime that with respect to the argument.
This is called the argument in that case.
Derivative of natural log is 1 over what?
The argument.
And you cover up natural log with your hand,
and you keep going.
And you say, next I go, times the derivative
of this square, plus 1, prime with respect to t.
So I go times 2t.
And that's what we have.
And they say, when you explain it like that, they said to me,
I can understand it.
But I'm having a problem understanding it
when you express this diagram-- that it throws me off.
So in order to avoid that kind of theoretical misconception,
I'm saying, let us see what the heck this is.
d dt of g of f of t, because this is what you're doing,
has to have some understanding.
The problem is that Mister f of t, that lives here,
has a different argument.
The letter in B should be, let's say, u.
That doesn't say anything practically.
How do you differentiate with respect to what?
You cannot say d dt here.
So you have to call f of t something generic.
You have to have a generic variable for that.
So you have then dg du, at what specific value of u?
At the specific value of u that we have as f of t.
Do you understand the specificity of this?
Times-- that's the chain rule, the product coming
from the chain rule-- df pt.
You take du dt or d of dt.
It is the same thing.
Say it again, df dt.
I had a student ask me, what if I put du dt?
Would it be wrong?
No, as long as you understand that u is a-something,
as the image of this t.
Do you know what he liked?
He said, do you know what I like about that?
I like that I can imagine that these are two cowboys-- I
told the same thing to my son.
He was so excited, not about that,
but about these two cowboys.
Of course, he is 10.
These are the cowboys.
They are across.
One is on top of the building there,
shooting at this guy, who is here
across the street on the bottom.
So they are annihilating each other.
They shoot and they die.
And they die, and you're left with 1/3.
The same idea is that, actually, these guys do not simplify.
du and-- [? du, ?] they're not cowboys who shoot at each other
at the same time and both die at the same time.
It is not so romantic.
But the idea of remembering this formula is the same.
Because practically, if you want to annihilate the two cowboys
and put your hands over them so you don't see them anymore,
du dt, you would have to remember, oh,
so that was the derivative with respect
to t that I initially have of the guy on top,
which was g of f of the composed function.
So if you view g of f of t as the composed function,
who is that?
The composition g composed with f of t
is the function g of f of t.
This is the function that you want to differentiate
with respect to time, t.
This is this, prime with respect to t.
It's like they would be killing each other, and you would die.
And I liked this idea, and I said,
I should tell that to my students and to my son.
And, of course, my son started jumping around
and said that he understands multiplication of fractions
better now.
They don't learn about simplifications-- I don't
know how they teach these kids.
It became so complicated.
It's as if mathematics-- mathematics is the same.
It hasn't changed.
It's the people who make the rules
on how to teach it that change.
So he simply doesn't see that this simplifies.
And when I tell him simplify, he's like, what is simplify?
What is this word simplify?
My teacher doesn't use it.
So I feel like sometimes I want to shoot myself.
But he went over that and he understood about the idea
of simplification.
[? He ?] composing something on top
and the bottom finding the common factors up and down,
crossing them out, and so on.
And so now he knows what it means.
But imagine going to college without having
this early knowledge.
You come to college, you were good in school,
and you've never learned enough simplification.
And then somebody like me, and tells you simplification.
You say, she is a foreigner.
She has a language barrier that is [INAUDIBLE] she has
that I've never heard before.
So I wish the people who really re-conceive, re-write
the curriculum for K12 would be a little bit
more respectful of the history.
Imagine that I would teach calculus
without ever telling you anything about Leibniz, who
was Leibniz, he doesn't exist.
Or Euler, or one of these fathers.
They are the ones who created these notations.
And if we never tell you about them, that I guess,
wherever they are, it is an injustice that we are doing.
All right.
Chain rule in Chapter 5 of Calc 3.
This is a little bit more complicated,
but I'm going to teach it to you because I like it.
Imagine that you have z equals x squared plus y squared.
What is that?
It's an example of a graph.
And I just taught you what a graph is.
But imagine that xy follow a curve.
[INAUDIBLE] with respect to time.
And you will say, Magdalena, can you draw that?
What in the world do you mean that x and y follow a curve?
I'll try to draw it.
First of all, you are on a walk.
You are in a beautiful valley.
It's not a vase.
It's a circular paraboloid, as an example.
It's like an egg shell.
You have a curve on that.
You draw that.
You have nothing better to do than decorating eggs
for Easter.
Hey, wait.
Easter is far, far away.
But let's say you want to decorate eggs for Easter.
You take some color of paint and put paint on the egg.
You are actually describing an arc of a curve.
And x and y, their projection on the floor
will be x of t, y of t.
Because you paint in time.
You paint in time.
You describe this in time.
Now, if x of ty of t is being projected on the floor.
Of course, you have a curve here as well, which is what?
Which it will be x of t, y of t, z of t.
Oh, my god.
Yes, because the altitude also depends on the motion in time.
All right.
So what's missing here?
It's missing the third coordinate, duh, that's
0 because I'm on the floor.
I'm on the xy plane, which is the floor z equals z.
But now let's suppose that I want
to say this is f of x and y, and I want to differentiate
f with respect to t.
And you go, say what?
Oh, my god.
What is that?
I differentiate f with respect to t.
By differentiating f with respect to t,
I mean that I have f of x and y differentiated
with respect to t.
And you say, wait, Magdalena.
This doesn't make any sense.
And you would be right to say it doesn't make any sense.
Can somebody tell me why it doesn't make any sense?
It's not clear where in the world the variable t is inside.
So I'm going to say, OK, x are themselves functions
of t, functions of that.
x of t, y of t.
If I don't do that, it's not clear.
So this is a composed function just like this one.
Look at the similarity.
It's really beautiful.
This is a function of a function, g of f.
This is a function of two functions.
Say it again, f is a function of two functions, x and y.
This was a function of a function of t.
This was a function of two functions of t.
Oh, my God.
How do we compute this?
There is a rule.
It can be proved.
We will look a little bit into the theoretical justification
of this proof later.
But practically what you do, you say,
I have to have some order in my life.
OK.?
So the way we do that, we differentiate first
with respect to the first location, which is x.
I go there, but I cannot write df dx because f is a mother
of two babies.
f is a function of two variables, x and y.
She has to be a mother to both of them;
otherwise, they get jealous of one another.
So I have to say, partial of f with respect to x,
I cannot use d.
Like Leibniz, I have to use del, d of dx.
At the point x of dy of t, this is the location I have.
Times what?
I keep derivation.
I keep derivating, like don't drink and derive.
What is that?
The chain rule.
Prime again, this guy x with respect to t, dx dt.
And then you go, plus because she has
to be a mother to both kids.
The same thing for the second child.
So you go, the derivative of f with respect
to y, add x of ty of t times dy dt.
So you see on the surface, x and y are moving according to time.
And somehow we want to measure the derivative
of the resulting function, or composition function,
with respect to time.
This is a very important chain rule
that I would like you to memorize.
A chain rule.
Chain Rule No.
1.
Is it hard?
No, but for me it was.
When I was 21 and I saw that-- and, of course,
my teacher was good.
And he told me, Magdalena, imagine that instead of del you
would have d's.
So you have d and d and d and d.
The dx dx here, dy dy here, they should be in your mind.
They are facing each other.
They are across on a diagonal.
And then, of course, I didn't tell my teacher my idea
with the cowboys, but it was funny.
So this is the chain rule that re-makes, or generalizes
this idea to two variables.
Let's finish the example because we didn't do it.
What is the derivative of f in our case?
df dt will be-- oh, my god-- at any point p, how arbitary,
would be what?
First, you write with respect to x.
2x, right?
2x.
But then you have to compute this dx, add the pair you give.
And the pair they gave you has a t.
So 2x is add x of ty-- if you're going
to write it first like that, you're
going to find it weird-- times, I'm done with the first guy.
Then I'm going to take the second guy in red,
and I'll put it here.
dx dt, but dx dt everybody knows.
[INAUDIBLE] Let me write it like this.
Plus [INAUDIBLE] that guy again with green-- dy
computed at the pair x of dy of [? t ?] times,
again, in red, dy dt.
So how do we write the whole thing?
Could I have written it from the beginning better?
Yeah.
2x of t, dx dt plus 2y of t dy.
Is it hard?
No, this is the idea.
Let's have something more specific.
I'm going to erase the whole thing.
I'll give you a problem that we gave on the final
a few years ago.
And I'll show you how my students cheated on that.
And I let them cheat, in a way, because in the end
they were smart.
It didn't matter how they did the problem, as long as they
got the correct answer.
So the problem was like that.
And my colleague did that many years ago, several years ago,
did that several times.
So he said, let's do f of t, dt squared and g of t.
I'll I'll do this one, dq plus 1.
And then let's [INAUDIBLE] the w of u
and B, exactly the same thing I gave you before, [INAUDIBLE] I
remember that.
And he said, compute the derivative of w of f of t,
and g of t with respect to t.
And you will ask, wait a minute here.
Why do you put d and not del?
Because this is a composed function that in the end
is a function of t only.
So if you do it as a composed function,
because this goes like this.
t goes to two functions, f of t and u.
And there is a function w that takes both of them, that
is a function of both of them.
In the end, this composition that's straight from here
to here, is a function of one variable only.
So my students then-- it was in the beginning of the examine,
I remember.
And they said, well, I forgot, they said.
I stayed up almost all night.
Don't do that.
Don't do what they did.
Many of my students stay up all night
before the final because I think I scare people,
and that's not what I mean.
I just want you to study.
But they stay up before the final and the next day,
I'm a vegetable.
I don't even remember the chain rule.
So they did not remember the chain rule
that I've just wrote.
And they said, oh, but I think I know how to do it.
And I said, shh.
Just don't say anything.
Let me show you how the course coordinator wanted
that done several years ago.
So he wanted it done by the chain rule.
He didn't say how you do it.
OK?
He said just get to the right answer.
It doesn't matter.
He wanted it done like that.
He said, dw of f of tg of p with respect to t,
would be dw du, instead of u you have f of t.
f of tg of t times df dt plus dw with respect
to the second variable.
So this would be u, and this would be v with respect
to the variable v, the second variable
where [? measure ?] that f of dg of t.
Evaluate it there times dg dt.
So it's like dv dt, which is dg dt. [INAUDIBLE] So he did that,
and he expected people to do what?
He expected people to take a u squared the same 2 times
u, just like you did before, 2 times.
And instead of u, since u is f of t to [INAUDIBLE] puts
2f of t, this is the first squiggly thing, times v of dt.
2t is this smiley face.
This is 2t plus-- what is the f dv?
Dw with respect to dv is going to be 2v 2 time gf t.
When I evaluate add gf t, this funny fellow
with this funny fellow, times qg d, which, with your permission
I'm going to erase and write 3p squared.
And the last row he expected my students to write
was 2t squared times 2t plus 2pq plus 1, times 3t squared.
Are you guys with me?
So [INAUDIBLE] 2t 2x 2t squared, correct.
I forgot to identify this as that.
All right.
So in the end, the answer is a simplified answer.
Can you tell me what it is?
I'm too lazy to write it down.
You compute it.
How much is it simplified?
Find it as a polynomial.
STUDENT: [INAUDIBLE].
PROFESSOR TODA: So you have 6, 6--
STUDENT: 16 cubed plus 3--
PROFESSOR TODA: T to the 5th plus--
STUDENT: [INAUDIBLE].
PROFESSOR TODA: In order, in order.
What's the next guy?
STUDENT: [INAUDIBLE].
PROFESSOR TODA: 4t cubed.
And the last guy--
STUDENT: 6t squared.
PROFESSOR TODA: 6t squared.
Yes?
Did you get the same thing?
OK.
Now, how did my students do it?
[INAUDIBLE]
Did they apply the chain rule?
No.
They said OK, this is how it goes.
W of U of T and V of T is U is F. So this guy is T squared,
T squared squared, plus this guy is T
cubed plus 1 taken and shaken and squared.
And then when I do the whole thing, derivative
of this with respect to T, I get--
I'm too lazy-- T to the 4 prime is 40 cubed.
I'm not going to do on the map.
2 out T cubed plus 1 times chain rule, 3t squared.
40 cubed plus 16 to the 5 plus-- [INAUDIBLE] 2 and 6t squared.
So you realize that I have to give them 100%.
Although they were very honest and said, we blanked.
We don't remember the chain rule.
We don't remember the formula.
So that's fine.
Do whatever you can.
So I gave them 100% for that.
But realize that the author of the problem
was a little bit naive.
Because you could have done this differently.
I mean if you wanted to actually test the whole thing,
you wouldn't have given-- let's say you wouldn't have given
the actual-- yeah, you wouldn't have given the actual functions
and say write the chain formula symbolically
for this function applied for F of T and G of T.
So it was-- they were just lucky.
Remember that you need to know this chain rule.
It's going to be one of the problems
to be emphasized in the exams.
Maybe one of the top 15 or 16 most important topics.
Is that OK?
Can I erase the whole thing?
OK.
Let me erase the whole thing.
OK.
Any other questions?
No?
I'm not going to let you go right away,
we're going to work one more problem or two more
simple problems.
And then we are going to go.
OK?
So question.
A question.
What do you think the gradient is good at?
Two reasons, right.
Review number one.
If you have an increasingly defined function,
then the gradient of F was what?
Equals direction of the normal to the surface S--
let's say S is given increasingly at the point
with [INAUDIBLE].
But any other reason?
Let's take that again.
Z equals x squared plus y squared.
Let's compute a few partial derivatives.
Let's compute the gradient.
The gradient is Fs of x, Fs of y, where this is F of xy
or Fs of xi plus Fs of yj.
[INAUDIBLE]
And we drew it.
I drew this case, and we also drew another related example,
where we took Z equals 1 minus x squared minus y squared.
And we went skiing.
And we were so happy last week to go skiing, because we still
had snow in New Mexico, and we-- and we
said now we computed the Z to be minus 2x minus 2y.
And we said, I'm looking at the slopes.
This is the x duration and the y duration.
And I'm looking at the slopes of the lines of these two curves.
So one that goes down, like that.
So this was for what?
For y equals 0.
And this was for x equals 0.
Curve, x equals 0 curve in plane.
Right?
We just cross-section our surface,
and we have this [INAUDIBLE].
And then we have the two tangents, two slopes.
And we computed them everywhere.
At every point.
But realize that to go up or down these hills,
I can go on a curve like that, or I
can go-- remember the train of Mickey Mouse going
on the hilly point on the hill?
We try to take different paths.
We are going hiking.
We are going hiking, and we'll take hiking through the pass.
OK.
How do we get the maximum rate of change of the function
Z equals F of x1?
So now I'm anticipating something.
I'd like to see your intuition, your inborn sense of I
know what's going to happen.
And you know what that from Mister--
STUDENT: Heinrich.
PROFESSOR TODA: [? Heinrich ?] from high school.
So I'm asking-- let me rephrase the question
like a non-mathematician.
Let's go hiking.
This is [INAUDIBLE] we go to the lighthouse.
Which path shall I take on my mountain, my hill,
my god knows what geography, in order
to obtain the maximum rate of change?
That means the highest derivative.
In what direction do I get the highest derivative?
STUDENT: In what direction you get the highest derivative--
PROFESSOR TODA: So in which direction--
in which direction on this hill do
I get the highest derivative?
The highest rate of change.
Rate of change means I want to get the fastest possible way
somewhere.
STUDENT: The shortest slope?
Along just the straight line up.
PROFESSOR TODA: Along--
STUDENT: You don't want to take any [INAUDIBLE].
PROFESSOR TODA: Right.
STUDENT: [INAUDIBLE].
It could be along any axis.
PROFESSOR TODA: So could you see which direction
those are-- very good.
Actually you were getting to the same direction.
So [INAUDIBLE] says Magdalena, don't be silly.
The actual maximum rate of change for the function Z
is obviously, because it is common sense,
it's obviously happening if you take the so-called-- what
are these guys?
[INAUDIBLE], not meridians.
STUDENT: Longtitudes?
PROFESSOR TODA: OK.
That is-- OK.
Suppose that we don't hike, because it's too tiring.
We go down from the top of the hill.
Ah, there's also very good idea.
So when you let yourself go down on a sleigh,
don't think bobsled or anything-- just a sleigh,
think of a child's sleigh.
No, take a plastic bag and put your butt in it
and let yourself go.
What is their direction actually?
Your body will find the fastest way to get down.
The fastest way to get down will happen exactly
in the same directions going down
in the directions of these meridians.
OK?
And now, [INAUDIBLE].
The maximum rate of change will always
happen in the direction of the gradient.
You can get a little bit ahead of time
by just-- I would like this to [INAUDIBLE] in your heads
until we get to that section.
In one section we will be there.
We also-- it's also reformulated as the highest, the steepest,
ascent or descent.
The steepest.
The steepest ascent or the steepest descent
always happens in the direction of the gradient.
Ascent is when you hike to the top of the hill.
Descent is when you let yourself go in the plastic [INAUDIBLE]
bag in the snow.
Right?
Can you verify this happens just on this example?
It's true in general, for any smooth function.
Our smooth function is a really nice function.
So what is the gradient?
Well again, it was 2x 2y, right?
And that means at a certain point, x0 y0, whenever you are,
guys you don't necessarily have to start
from the top of the hill.
You can be-- OK, this is your cabin.
And here you are with friends, or with mom and dad,
or whoever, on the hill.
You get out, you take the sleigh, and you go down.
So no matter where you are, there you go.
You have 2x0 times i plus 2y0 times j.
And the direction of the gradient will be 2x0 2y0.
Do you like this one?
Well in this case, if you were-- suppose
you were at the point [INAUDIBLE].
You are at the point of coordinates--
do you want to be here?
You want to be here, right?
So we've done that before.
I'll take it as 1 over [? square root of ?]
2-- I'm trying to be creative today-- [INAUDIBLE] y equals 0,
and Z equals-- what's left?
1/2, right?
Where am I?
Guys, do you realize where I am?
I'll [? take a ?] [INAUDIBLE].
y0.
So I need to be on this meridian on the red thingy.
And somewhere here.
What's the duration of the gradient here?
Delta z at this p.
Then you say ah, well, I don't get it.
I have-- the second guy will become 0, because y0 is 0.
The first guy will become 1 over square root of 2.
So I have 2 times 1 over square root of 2 times i plus 0j.
It means in the direction of i-- in the direction of i-- from p,
I have the fastest-- fastest, Magdalena, fastest-- descent
possible.
But we don't say in the direction of i
in our everyday life, right?
Let's say geographic points.
We are-- I'm a bug, and this is north.
This is south.
This is east.
And this is west.
So if I go east, going east means going in the direction i.
Now suppose-- I'm going to finish with this one.
Suppose that my house is not on the prairie
but my house is here.
House, h.
Find me a wood point to be there.
STUDENT: Northeast.
Or to get further down.
PROFESSOR TODA: Anything, what would look like why I'm here?
x0, y0, z0.
Hm.
1/2, 1/2, and I need the minimum.
So I want to be on the bisecting plane between the two.
You understand?
This is my quarter.
And I want to be in this bisecting plane.
So I'll take 1/2, 1/2, and what results from here?
I have to do math.
1 minus 1/4 minus 1/4 is 1/2.
Right?
1/2, 1/2, 1/2.
This is where my house is [? and so on. ?]
And this is full of smoke.
And what is the maximum rate of change?
What is the steepest descent is the trajectory
that my body will take when I let myself go down
on the sleigh.
How do I compute that?
I will just do the same thing.
Delta z at the point x0 equals 1/2, y0 equals 1/2,
z0 equals 1/2.
Well what do I get as direction?
That will be the direction of the gradient.
2 times 1/2-- you guys with me still?
i plus 2 times 1/2 with j.
And there is no Mr. z0 In the picture.
Why?
Because that will give me the direction
like on-- in a geographic way.
North, west, east, south.
These are the direction in plane.
I'm not talking directions on the hill,
I'm talking directions on the map.
These are directions on the map.
So what is the direction i plus j on the map?
If you show this to a geography major and say,
I'm going in the direction i plus j on the map,
he will say you are crazy.
He doesn't understand the thing.
But you know what you mean.
East for you is the direction of i in the x-axis.
[INAUDIBLE]
And this is north.
Are you guys with me?
The y direction is north.
So I'm going perfectly northeast at a 45-degree angle.
If I tell the geography major I'm
going northeast perfectly in the middle, he will say I know.
But you will know that for you, that is i plus j.
Because you are the mathematician.
Right?
So you go down.
And this is where you are.
And you're on the meridian.
This is the direction i plus j.
So if I want to project my trajectory-- I went down
with the sleigh, all the way down-- project the trajectory,
my trajectory is a body on the snow.
Projecting it on the ground is this one.
So it is exactly the direction i plus j.
Right, guys?
So exactly northeast perfectly at 45-degree angles.
Now one caveat.
One caveat, because when we get there,
you should be ready already, in 11.6 and 11.7.
When we will say direction, we are also crazy people.
I told you, mathematicians are not normal.
You have to be a little bit crazy
to want to do all the stuff in your head like that.
i plus j for us is not a direction most of the time.
When we say direction, we mean we normalize that direction.
We take the unit vector, which is unique,
for responding to i plus j.
So what is that unique unit vector?
You learned in Chapter 9 everything is connected.
It's a big circle.
i plus j, very good.
So direction is a unit vector for most mathematicians,
which means you will be i plus j over square root of 2.
So in Chapter 5, please remember, unlike Chapter 9,
direction is a unit vector.
In Chapter 9, Chapter 10, it said direction lmn,
direction god knows what.
But in Chapter 11, direction is a vector in plane,
like this one, i plus [INAUDIBLE]
has to be a unique normal-- a unique vector.
OK?
And we-- keep that in mind.
Next time, when we meet on Thursday,
you will understand why we need to normalize it.
Now can we say goodbye to the snow and everything?
It's not going to show up much anymore.
Remember this example.
But we will start with flowers next time.
OK.
Have a nice day.
Yes, sir?
Let me stop the video.