– [Teacher] So, just as a

reminder of where we are, we’ve got this very

non-linear transformation and we showed that if you zoom

in on a specific point while that transformation is happening, it looks a lot like something linear and we reason that you can figure out what linear transformation that looks like by taking the partial derivatives

of your given function, the one that I defined up here, and then turning that into a matrix. And what I want to do here

is basically just finish up what I was talking about by computing all of those partial derivatives. So, first of all, let me

just rewrite the function back on the screen so we have it in a convenient place to look at. The first component is x plus sin of y. Sin of y and then y plus sin of x was the second component. So, what I want to do

here is just compute all of those partial derivatives to show what kind of thing this looks like. So, let’s go ahead and

get rid of this word and I’ll go ahead and kind

of redraw the matrix here. So, for that upper left component, we’re taking the partial derivative with respect to x of the first component. So, we look up at this first component and the partial derivative

with respect to x is just one. Since there’s one times x plus

something that has nothing to do with x and then below that, we take the partial derivative of the second component

with respect to x down here. And that guy, the y, well

that looks like a constant so nothing happens, and

the derivative of sin of x becomes cosine of x. And then up here, we’re

taking the partial derivative with respect to y of the first component; that upper one here, and for that, partial derivative of x,

with respect to y, is zero and partial derivative of sin of y, with respect to y, is cosine of y. And then, finally, the partial derivative of the second component with

respect to y looks like one because it’s just one

times y plus some constant. And this is the general Jacobian

as a function of x and y, but if we want to understand what happens around this specific

point that started off at, well, I think I recorded

it here at negative two, one, we plug that in to each

one of these values. So, when we plug in negative two, one. So go ahead and just

kind of again, rewrite it to remember we’re plugging

in negative two, one as our specific point,

that matrix as a function, kind of a matrix valued function, becomes one, and then next we have cosine, but we’re plugging in negative two for x, cosine of negative two,

and if you’re curious, that is approximately equal to, I calculated this earlier. Negative zero point four

two, if you just want to think in terms of a number there. Then for the upper right,

we have cosine again, but now we’re plugging in the value for y, which is one and cosine of

one is approximately equal to zero point five four;

and then bottom right, that’s just another constant: one. So, that is the matrix, just

as a matrix full of numbers, and just as kind of a gut

check we can take a look at the linear transformation

this was supposed to look like, and notice how the first basis factor, the thing it got turned into,

which is this factor here, does look like it has coordinates one and negative zero point four two, right? It’s got this rightward component that’s about as long as

the vector itself started and then this downward component, which I think that’s pretty believable that that’s negative zero point four two. And then, likewise, this

second column is telling us what happened to that second basis factor, which is the one that looks like this. And again, its y component

is about as long as how it started, right, the length of one. And then the rightward component

is around half of that, and we actually see that in the diagram, but this is something you compute. Again, it’s pretty straightforward. You just take all of the

possible partial derivatives, and you organize them

into a grid like this. So, with that, I’ll see

you guys next video.

## 12 Comments

## Ella noelle

First lol jk

## Donderboor N

WTF did I come across, i am only 13

## Educational Videos for Students (Cartoons on Bullying, Leadership & More)

this is excellent

## Chicken Strangler

Clicked to get my mid blown, it was great.

## bloodspilla55

what the fuck

## Tristan Batchler

I wish I had this when we did variable transformations last semester 🙁

## meme

So if you zoom in on ANY function is it locally linear? Or just special functions? If only certain ones are locally linear does the Jacobian have any use for them?

## Aaron Gershman

you're awesome :')

## sofisticated ranchbroh420

Does jacobian and hessians only apply to functions going fron R^n space to R^n space?? I first learned Jacobian with regard to R^2 to R^1 space, like traditional 3d functions but this exapmle and general explanations seem to be talking about transformations*** where we have the in with the same dimensionality as the out. Comments? Please!

## X B

Is this 3Blue1Brown?

## Ray Broomall

Pre Calc at 13 is dope… don't get bent. If you have any idea of Linear Algebra then this isn't too difficult. But with a full load of Pre Calc you may not have time for this. If nothing else, this is telling you why you need Pre Calc . If you can visualize what he did here the two points that you can take away is that you can plug in any two number pairs into the last matrix (cos) and it will spit out how that point in the XY plane has been changed by the function. And that any non linear function can be made to look linear if you take a close enough look. (the limit of the function) It's hard work but you are doing it, and that's cool.

## Abdullah AboMuhammad

where is next vidoe?