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CHANGE OF VARIABLES

Essentials

Main ideas

  • There are many ways to solve this problem!
  • Using Jacobians (and inverse Jacobians)

Prerequisites

  • Surface integrals
  • Jacobians
  • Green's/Stokes' Theorem

Warmup

Perhaps a discussion of single and double integral techniques for solving this problem.

Props

  • whiteboards and pens

Wrapup

This is a good conclusion to the course, as it reviews many integration techniques. We emphasize that (2-dimensional) change-of-variable problems are a special case of surface integrals.

Here are some of the methods one could use to do these integrals:

  • change of variables (at least 2 ways)
  • Area Corollary to Green's Theorem (at least 2 ways)
  • ordinary single integral (at least 2 ways)
  • ordinary double integral (at least 2 ways)
  • surface integral

Details

In the Classroom

  • Some students will want to simply use Jacobian formulas; encourage such students to try to solve this problem both by computing $\Partial{(x,y)}{(u,v)}$ and by computing $\Partial{(u,v)}{(x,y)}$.
  • Other students will want to work directly with $d\rr_1$ and $d\rr_2$. This works fine if one first solves for $x$ and $y$ in terms of $u$ and $v$.
  • Students who compute $d\rr_1$ and $d\rr_2$ directly can easily get confused, since they may try to eliminate $x$ or $y$, rather than $u$ or $v$. 1) Emphasize that one must choose parameters, both on the region, and on each curve, and that $u$ and $v$ are chosen to make the limits easy.

Subsidiary ideas

  • Review of Green's Theorem
  • Review of single integral techniques
  • Review of double integral techniques

Homework

(none yet)

Essay questions

(none yet)

Enrichment

  • Discuss the 3-dimensional case, perhaps relating it to volume integrals.

1) Along the curve $v=\hbox{constant}$, one has $dy=v\,dx$, so that $d\rr_1 = dx\,\ii + dy\,\jj = (\ii + v\,\jj)\,dx$, which some students will want to write in terms of $x$ alone. But one needs to express this in terms of $du$! This can be done using $du = x\,dy + y\,dx = x (v\,dx) + y\,dx = 2y\,dx$, so that $d\rr_1 = (\ii + v\,\jj) \,\frac{du}{2y}$. A similar argument leads to $d\rr_2 = (-\frac{1}{v}\,\ii+\jj)\,\frac{x\,dv}{2}$ for $u=\hbox{constant}$, so that $d\SS = d\rr_1\times d\rr_2 = \kk \,\frac{x}{2y}\,du\,dv = \kk \,{du\,dv\over2v}$. This calculation can be done without solving for $x$ and $y$, provided one recognizes $v$ in the penultimate expression.

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