15.9 Videos Guide

15.9a

Exercise:

• Find the image of the set $S$ under the given transformation.
  $S$ is the square bounded by the lines $u=0$, $u=1$, $v=0$, $v=1$;
  $x=v$, $y=u\left(1+v^{2}\right)$

15.9b

• The Jacobian

  • For a transformation $T$ that maps S in the $uv$-plane onto $R$ in the $xy$-plane with inverse transformation $T^{-1}$ (which maps $R$ onto S), the Jacobian of $x$ and $y$ with respect to $u$ and $v$ is

$$\frac{\partial(x,\ y)}{\partial(u,\ v)}=\left|\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{matrix}\right|$$

• Change of variables for double integrals (under $T$)

  • $\iint^{\ }_{R}{f(x,\ y)\ dA=\iint^{\ }_{S}{f\left(g(u,\ v),h(u,\ v)\right)}}\left|\frac{\partial(x,\ y)}{\partial(u,\ v)}\right|\ dudv$

Exercises:

15.9c

• Use the given transformation to evaluate the integral.
  $\iint^{\ }_{R}{(4x+8y)\ dA}$, where $R$ is the parallelogram with vertices $(-1,\ 3)$, $(1,\ -3)$, $(3,\ -1)$, and $(1,\ 5)$;
  $x=\frac{1}{4}(u+v)$, $y=\frac{1}{4}(u-3v)$

15.9d

• A region $R$ in the $xy$-plane is given. Find equations for a transformation $T$ that maps a rectangular region $S$ in the $uv$-plane onto $R$, where the sides of $S$ are parallel to the $u$- and $v$-axes.
  $R$ is bounded by the hyperbolas $y=\frac{1}{x}$, $y=\frac{4}{x}$ and the lines $y=x$, $y=4x$ in the first quadrant

15.9e

• Evaluate the integral by making an appropriate change of variables.
  $\iint^{\ }_{R}{(x=y)e^{x^{2}-y^{2}}\ dA}$, where $R$ is the rectangle enclosed by the lines $x-y=0$,
  $x-y=2$, $x+y=0$, and $x+y=3$

• Using $T:S\rightarrow R$ and $T^{-1}:\ R\rightarrow S$

  • $u=f(x,\ y)$ and $v=g(x,\ y)$ are useful in finding limits of integration

  • $x=x(u,\ v)$ and $y=y(u,\ v)$ are useful for finding $\frac{\partial(x,\ y)}{\partial(u,\ v)}$

  • Substitute using whichever is most convenient