Multivariable calculus, class # 24: change of variables

Mathematician spotlight: Colin Adams

We briefly discuss knot invariants and the petal number. We show the amazing trick to compute the area under the bell curve (Gaussian distribution) by squaring the area and turning it into a double integral, then converting to polar coordinates. We give the formula for the Jacobian area factor for a (u,v) substitution in general, then show that it is a generalization of u-substitutions from single-variable calculus. We do two examples of (u,v) substitution, a linear one (where we can solve for x and y in terms of u and v) and a nonlinear one (where we use the inverse of the determinant of the inverse transformation matrix).

[chasemckinley]

Really appreciated the perspective given on how the Jacobian determinant relates to the one dimensional u substitutions from earlier calculus....I've seen visual computer generated pics/visual aids that helped a good bit, but this video gave substnace to where those random multicolored blobs came from/what they were doing haha

Thanks so much for doing more than writing down webassign course login info and leaving it to us from there!

[lionhawk]

Diana, you are a remarkable teacher. Yesterday I woke up as just another amateur maths student, and today I find myself the Thomas T. Read Professor of Mathemeatics at Williams College - a place I have never even been to. Remarkable.

[wallyduboss]

There is another way using triple integrals. I wasn't aware of this approach.

[nealfurman]

First of all, thank you so much for all of this! you do it so well.. what a great gift for humanity.
My I ask you a question? about your drawing.. when you describe the change of variables at about 24:00, could these shapes really represent a linear transformation? I can't imagine how the grid lines would stay parallel and evenly spaced in that case, is that theoretically possible for such rounded undefined shapes to represent a linear transformation? would it be like to squeez or to stretch (symetrically) a piece of plasticine?, on the other hand, if I make a hole through the plasticine with my finger it wouldn't be a linear transformation (or should it be called a deformation) and would be irreversible (unless there's a special function that could somehow restore it to its' original form). Is my understanding at all reasonable? THANK YOU @Diana Davis

[KSadan]

Can I get the recording of the Kitao lecture ?? Thank you for the lectures.