Lec 18: Change of variables | MIT 18.02 Multivariable Calculus, Fall 2007

Lecture 18: Change of variables.
View the complete course at: ocw.mit.edu/18-02SCF10
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[@samiulbasirtasin40]

Lecture 1: Dot Product Lecture 2: Determinants Lecture 3: Matrices Lecture 4: Square Systems Lecture 5: Parametric Equations Lecture 6: Kepler's Second Law Lecture 7: Exam Review (goes over practice exam 1a at 24 min 40 seconds) Lecture 8: Partial Derivatives Lecture 9: Max-Min and Least Squares Lecture 10: Second Derivative Test Lecture 11: Chain Rule Lecture 12: Gradient Lecture 13: Lagrange Multipliers Lecture 14: Non-Independent Variables Lecture 15: Partial Differential Equations Lecture 16: Double Integrals Lecture 17: Polar Coordinates Lecture 18: Change of Variables Lecture 19: Vector Fields Lecture 20: Path Independence Lecture 21: Gradient Fields Lecture 22: Green's Theorem Lecture 23: Flux Lecture 24: Simply Connected Regions Lecture 25: Triple Integrals Lecture 26: Spherical Coordinates Lecture 27: Vector Fields in 3D Lecture 28: Divergence Theorem Lecture 29: Divergence Theorem (cont.) Lecture 30: Line Integrals Lecture 31: Stokes' Theorem Lecture 32: Stokes' Theorem (cont.) Lecture 33: Maxwell's Equations Lecture 34: Final Review Lecture 35: Final Review (cont.)

[@harendrasingh2319]

Thanks a lot 😘

[@umakantmishra6139]

Where is jacobians

[@rahuldhungel]

Lecture 18 : Change Of Variables and The Jacobian

[@shaidniloy5356]

Thanks bro, saved some time.

[@DemiHalf]

yo no entiendo; was this info not in the playlist?

[@LAnonHubbard]

It was awesome to learn about the Jacobian. I've seen this here and there and never understood what it was. Thanks!!

[@HenryEBrass]

Dr. Auroux is terrific at impromptu sketching. 19:14, look at the neatness of all six boards!

[@wontpower]

At first, I though "rangle" was just a weird shorthand for rectangle

[@athelstanrex]

Lmao

[@BrownMunde606]

Me too

[@wlockhart]

So did I. However these students burst into laughter if the wind blows.

[@avixek]

I am completely following the mit courseware to study my pre engineering courses awesome > thanks MIT

[@ashwinjain5566]

i was completely lost for 2 whole days trying to make intuitive sense of it after my college professor taught us this. I mean i only got taught the algorithm for finding the jacobian but not what it actually is but after watching this, it finally clicked and i am so happy lmao

[@NehadHirmiz]

Prof. Auroux, Thank you for these wonderful lectures

[@vinaydeepbeeram6698]

Professor Dennis Auroux's #1 Fan

[@yashagarwal3999]

The best ever way to teach Jacobians, wow greatest of the great, Jacobians .......great, the professor explanation ...great

[@wildfire5932]

17:29 by the way a rangle is a bit of gravel fed to a hawk

[@atul6147]

he does it not as if he's doing math, but like art, like poetry

[@prafullvilas1931]

I wouldn't have built this mentality that I'll never be able to get even average at maths if we had teachers like this 🙏🙏🙏

[@111abdurrahman]

This is a pure enjoyment to take Prof. Auorox lectures

[@norbitaspower]

great professor. thank you mit for this classes.

[@priyanksharma1124]

Nailed it! what an awesome lecture

[@serden8804]

Perfect professor and perfect explanation, how strange

[@leochen8149]

So brilliant lecture, help me a lot!

[@elamvaluthis7268]

Very nice.painstakingly explaining things nice.

[@saitrinath2591]

i like math series in ocw mit veryy much

[@GodlessNfree]

great video thanks

[@Manocheher.A]

I was looking at some of these videos as recently I took all my college math classes. I was able to follow along but got a bit out of tract. Later, I noticed I have yet to take this class and will need it in university. 😂

[@codingWorld709]

Thanks Sir. You are very handsom

[@binyillikcinar]

How does Jacobian go with nonlinear transformations? Namely if I have u=g(x,y) v=h(x,y) then will dudv=|J|dxdy be satisfied where J=(dg/dx dg/dy; dh/dx dh/dy) ? (here d's are actually referring to partial derivatives)

[@An1MuS]

Thank you a lot.

[@10keys]

@fermixx he did it without the drawing... he drew the bounds.... and the way he figured the second bound drawing he did was just plug in the x y points into the substitution equations he made for u v...

[@Pctech4uproductions]

Fantastic Lecture. Loved it.. Thanks a lot :)

[@shakesbeer00]

10:23 In the u-v coordinates, should the parallelogram still be referred to as Delta A, instead of Delta A'?

[@nadekang8198]

Again, I wish I had a calculus professor like him........instead of professors writing and teaching calculus using their own textbook purely with mathematica....it was so hard for me to get the intuition when I was in college...

[@rishavdhariwal4782]

Prof. Auroux used a linear approx for the general case to show how small changes in x and y can be related to u and v by their respective partial derivatives given that u=g(x,y) v=h(x,y), but this will only be valid for as said small changes in x and y. Therefore if changes in x and y are of finite value (meaning the region we are integrating over has some finite but non zero change in their x,y coords) then how will we relate the da element of such a region to the da' (area element in u,v coords)?

[@13Septem13]

@Djole0 "It is the second semester in the freshman calculus sequence"

[@shakesbeer00]

I am a little confused. It seems to me that u-axis and v-axis should not be perpendicular to each other.

[@DeepakSah3.0]

thanku

[@ZopteY]

THAAAAAAANKKKK YOUUUUU!

[@thatsfantastic313]

♥ Love to MIT

[@pdxginni]

@pdxginni Yes, it's just me. I had the screen maximized. Wouldn't pass without minimizing.

[@dishapanchal3060]

AT we reach 35 minutes of this lecture, Why they have written -rsin∅? what is x sub theta?

[@alexryyan]

I KNEW IT WAS A LINEAR TRANSFORMATION. I am retaking calc 3 cuz my HS credit didnt count and having taken lin alg, i saw the jacobian and im like WAITTTTT A SECOND. I cannot believe its not standard to teach what the Jacobian actually is (ik its not always linear but I digress)

[@quasirdp]

perfection

[@lee_land_y69]

intro to linear transformations and horrors of linear algebra haha

[@ishadev01]

thankyou sir

[@pyrole]

Finally understood Jacobian :)

[@debendragurung3033]

17:49 I like the word rangle

[@not_amanullah]

Thanks ❤🤍

[@bobkameron]

This course is awesome!

[@user-dt8xi7cd4e]

who didn't understand the transformation of variables I recommend you to watch wildberger linear algebra series on you tube it is very helpful, he will take step by step with a wonderful journey of linear algebra.

[@youmah25]

+rami nejem نعم

[@VineetKumar-wp9yr]

rami nejem go and watch 3blue1brown series on linear algebra

[@Souliee]

thanks :)

[@Bunk_Moreland]

what does "rangle" even mean? it is clearly a shorthand for rectangle

[@dagonmeister]

Get the feeling that classroom is full of trolling

[@fermixx]

is there any way to change the bounds of integrations without drawing? because most of the times it wont be so easy to draw. (or i wont have the time to do it) cant i just plug in the bound values into the change of variable's equations or do some other calculus?

[@SPRINGGREEN813]

I think the jacobian is used for that(btw, what are you doing after 14 years)

[@liviumircea6905]

Why du*dv = 5*dx*dy ? 'A' region is a rectangle but A' is a parallelogram and its area is not du*dv ...

[@jimallysonnevado3973]

I have some small issue about the Jacobian. Why do we even need to take the absolute value of the determinant? In single variable case, the derivative can similarly be thought of as the "exchange rate" for the dx to du if we change variables and in the sense similar to the Jacobian but in there even if the detivative is negative we don't take absolute values. But why should we take absolute values in the Jacobian determinant case?

[@joebrinson5040]

Because area is always a positive value and you are using the Jacobian as the "scaling factor" between two areas.

[@jimallysonnevado3973]

@@joebrinson5040 but that does not address the one dimensional case. For instance if you are going to compute the integral of x^2 dx from x=0 to x=1. If you want to do the substitution u=-x, you will similarly get du=-dx. The negative sign means we are multiplying by -1 which can also be thought of as the one-dimensional Jacobian. However, in this case we are not allowed to take absolute value because it will change the value of the integral. Ie, (if we take absolute values also in one-dimensional case) the integral in variable u will become integral (-u)^2 |-1| du from u=0 to u=-1. Which further becomes -1/3. But the actual value of the original integral is 1/3. Clearly, we don't take absolute values in one-dimensional case but why we do it in higher-dimensional integrals?

[@dsdsspp7130]

@@jimallysonnevado3973 I think you're right. you need to use the Jacobian itself both in the one dimension and higher dimensional cases. you could also just switch the lower and upper bounds with each other. if transformation of your space changes orientation then the integral will change signs so you need to multiply it by a negative one, regardless of dimension.

[@HanitpalSingh]

i didn't catch "rangle" until the students pointed it out

[@shoxruxturaev1931]

i'm confused to the extent that i don't know what i'm confused about. ( probably i'm missing something)

[@DestinyQx]

heh RANGLE! @ 17:15 i never took multi calc, he seems like a good prof

[@nate7645]

Filled with rage that my terrible calc 3 prof's lecture is a waste of time compared to this

[@shukiboy5514]

nice

[@ahmadhaitham6177]

How come he considered a hyperbola that passed by (x=0) and (y=0) simultaneously ??? at 46:55 .

[@hershyfishman2929]

The integral has to include all values of (x, y) from x = 0 to 1 and from y = 0 to 1. The actual point (0, 0) won't matter if you include it or not since the function is 0 at that point, but you can't start at the "next" point. There is no next point. As close as you get to (0,0) there will always be infinite points yet closer.

[@CotyAbadie]

The one person that disliked this didn't get into MIT.

[@novanecros9145]

Don't y'all love a good ol' rangle?

[@aniketkedare8]

Why where limit were not defined in terms of u and v in respect of x and y

[@sharifkhudairi5368]

sorry my sister dsnt know any of that :(

[@hnkul702]

16:31 "For any other rangle"?

[@9888565407]

hey supp/

[@jrkirby93]

rangle

[@crane8035]

I'm no expert but around 15:30 The prof. depicts the parallelogram in the UV plane ,but in reality the the sides are still on the axes in the UV plane because the axes them selves are sheared. But when we look at it from the vantage point of the XY plane ,they take the form such parallelogram.

[@squirtlesquirtle94]

Explanation of the relationship between the determinant of the Jacobian and the change of variables was handwavy at best! Would be nice to see a rigorous derivation of how these things work instead of accepting a magic formula...

[@abab7196]

40:32 hahauahaha he really hit the nail with saying some mysterious function ahhwuauaHWWUWHWHAA

[@pdxginni]

Is it just me, or does this lecture stop working at 2:31? I've restarted, but nothing.

[@psibarpsi]

Just you.

[@thereisnogodbutdalegribble5687]

Does anyone else really like the yellow chalk at 13:37 ?

[@chaitanyaanish8103]

my doubt in this lecture: sir told dA' is the area of the square in uv coordinates,but acc. to the diagram it dA' is the area of a parallelogram whose sides are not parallel to uv axes,how can that be possible

[@deeptochatterjee532]

Chaitanya Anish Actually he said the area of the square is proportional to dxdy and so when you convert the variables the scaling factor of the areas should be equal to the ratio of dudv to dxdy

[@rarulis]

rangle is a funny sounding word.

[@Djole0]

is this the first year of college?

[@psibarpsi]

Yes!

[@foundingtitan9759]

24:13

[@muddwell]

thats cool chalk.

[@sifiso5055]

Hagoromo Chalk

[@084148]

@MrDevin666 UCLA

[@weltschmerz137]

lol close up of rangle

[@MrDevin666]

MIT or UCLA for medicine?

[@DeadPool-jt1ci]

there not even comparable , i hope u chose MIT

[@MegaRickastley]

Those blackboards...

[@and1fer]

MATGRRIXUE!!

[@morani789]

Two people go to Harvard...

[@youmah25]

+morani789 and 3 to stanford

[@DieguezCreate]

i hear he was kermit's voice on sesame street for some time 0.o

[@MeAndCola]

I hate the camerawork

[@danideboe]

Prof Leonard is way better

[@kittycat1768]

ya, but his videos are insanely long (imagine how long it would take to watch an entire course)

[@malefetsanekoalane4549]

@@kittycat1768 If you think that is long try Prof Leonard. Now that is insane.

[@georgen9755]

Spam

[@MsKouider]

french is so hhhhh