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Lec 18: Change of variables | MIT 18.02 Multivariable Calculus, Fall 2007
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- čas přidán 6. 08. 2024
- Lecture 18: Change of variables.
View the complete course at: ocw.mit.edu/18-02SCF10
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
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.)
Thanks a lot 😘
Where is jacobians
Lecture 18 : Change Of Variables and The Jacobian
Thanks bro, saved some time.
yo no entiendo; was this info not in the playlist?
It was awesome to learn about the Jacobian. I've seen this here and there and never understood what it was. Thanks!!
Dr. Auroux is terrific at impromptu sketching. 19:14, look at the neatness of all six boards!
I am completely following the mit courseware to study my pre engineering courses awesome > thanks MIT
At first, I though "rangle" was just a weird shorthand for rectangle
Lmao
Me too
So did I. However these students burst into laughter if the wind blows.
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
Prof. Auroux, Thank you for these wonderful lectures
The best ever way to teach Jacobians, wow greatest of the great, Jacobians .......great, the professor explanation ...great
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 🙏🙏🙏
This is a pure enjoyment to take Prof. Auorox lectures
he does it not as if he's doing math, but like art, like poetry
great professor. thank you mit for this classes.
Professor Dennis Auroux's #1 Fan
17:29 by the way a rangle is a bit of gravel fed to a hawk
Nailed it! what an awesome lecture
So brilliant lecture, help me a lot!
Very nice.painstakingly explaining things nice.
Perfect professor and perfect explanation, how strange
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. 😂
Thanks Sir. You are very handsom
i like math series in ocw mit veryy much
great video thanks
Fantastic Lecture. Loved it.. Thanks a lot :)
Thank you a lot.
@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...
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...
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)
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)?
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...
@Djole0 "It is the second semester in the freshman calculus sequence"
THAAAAAAANKKKK YOUUUUU!
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)
10:23 In the u-v coordinates, should the parallelogram still be referred to as Delta A, instead of Delta A'?
I am a little confused. It seems to me that u-axis and v-axis should not be perpendicular to each other.
@pdxginni Yes, it's just me. I had the screen maximized. Wouldn't pass without minimizing.
♥ Love to MIT
thanku
This course is awesome!
Thanks ❤🤍
Finally understood Jacobian :)
perfection
AT we reach 35 minutes of this lecture,
Why they have written -rsin∅? what is x sub theta?
intro to linear transformations and horrors of linear algebra haha
17:49 I like the word rangle
thankyou sir
what does "rangle" even mean? it is clearly a shorthand for rectangle
Get the feeling that classroom is full of trolling
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.
+rami nejem نعم
rami nejem go and watch 3blue1brown series on linear algebra
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?
Because area is always a positive value and you are using the Jacobian as the "scaling factor" between two areas.
@@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?
@@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.
thanks :)
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?
I think the jacobian is used for that(btw, what are you doing after 14 years)
How come he considered a hyperbola that passed by (x=0) and (y=0) simultaneously ???
at 46:55 .
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.
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.
i didn't catch "rangle" until the students pointed it out
heh RANGLE! @ 17:15 i never took multi calc, he seems like a good prof
Why du*dv = 5*dx*dy ? 'A' region is a rectangle but A' is a parallelogram and its area is not du*dv ...
Why where limit were not defined in terms of u and v in respect of x and y
sorry my sister dsnt know any of that :(
i'm confused to the extent that i don't know what i'm confused about. ( probably i'm missing something)
40:32 hahauahaha he really hit the nail with saying some mysterious function ahhwuauaHWWUWHWHAA
Don't y'all love a good ol' rangle?
nice
16:31 "For any other rangle"?
hey supp/
rangle
Filled with rage that my terrible calc 3 prof's lecture is a waste of time compared to this
Does anyone else really like the yellow chalk at 13:37 ?
The one person that disliked this didn't get into MIT.
Is it just me, or does this lecture stop working at 2:31? I've restarted, but nothing.
Just you.
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
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
rangle is a funny sounding word.
thats cool chalk.
Hagoromo Chalk
MIT or UCLA for medicine?
there not even comparable , i hope u chose MIT
is this the first year of college?
Yes!
lol close up of rangle
Those blackboards...
24:13
@MrDevin666 UCLA
Two people go to Harvard...
+morani789 and 3 to stanford
MATGRRIXUE!!
i hear he was kermit's voice on sesame street for some time 0.o
Prof Leonard is way better
ya, but his videos are insanely long (imagine how long it would take to watch an entire course)
@@kittycat1768 If you think that is long try Prof Leonard. Now that is insane.
I hate the camerawork
Spam
french is so hhhhh