Page Description Equation
377 Jacobian of transformation $T:D\ast\subset\R^2\rightarrow\R^2$ \( T\left(u,v\right) = \left( x\left(u,v\right), y\left(u,v\right) \right) \) \[ \frac{\partial\left(x,y\right)} {\partial\left(u,v\right)} = \begin{vmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} &\frac{\partial y}{\partial v}\\ \end{vmatrix} \]
387 Jacobian of transformation $T:W\ast\subset\R^3\rightarrow\R^3,$
$T\left(u,v,w\right)$ \( =\left( x\left(u,v,w\right), y\left(u,v,w\right), z\left(u,v,w\right) \right) \) 		\[ \frac{\partial\left(x,y,z\right)} {\partial\left(u,v,w\right)} = \begin{vmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} &\frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} &\frac{\partial y}{\partial v} &\frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} &\frac{\partial z}{\partial v} &\frac{\partial z}{\partial w}\\ \end{vmatrix} \]
381 change of variables formula, one variable (aka substitution method) \[ \begin{multline*} \int_{I}f\left(x\right)\dx =\int_{I\ast} f\left(x\left(u\right)\right) \abslr{ \frac{dx}{du} }\du\\ =\int_{a}^{b} f\left(x\left(u\right)\right) \frac{dx}{du}\du =\int_{x\left(a\right)}^{x\left(b\right)} f\left(x\right)\dx \end{multline*} \]
382 change of variables formula, two variables $T:D\ast\rightarrow D$ is $C^1$ bijection, $T\left(u,v\right) = \left( x\left(u,v\right), y\left(u,v\right) \right),$ $f:D\rightarrow\R$ is any integrable function \[ \begin{multline*} \iint_{D}f\left(x,y\right)\dx\dy\\ =\iint_{D\ast} f\left( x\left(u,v\right), y\left(u,v\right) \right) \abslr{ \frac{\partial\left(x,y\right)} {\partial\left(u,v\right)} }\du\dv \end{multline*} \]
383 change of variables formula polar coordinates $T:D\ast\rightarrow D$ is $C^1$ bijection, $T\left(r,\theta\right)=\left(r\cos{\theta},r\sin{\theta}\right) =\left(x,y\right),$ $f:D\rightarrow\R$ is any integrable function \[ \begin{multline*} \iint_{D}f\left(x,y\right)\dx\dy\\ =\iint_{D\ast} f\left(r\cos{\theta},r\sin{\theta}\right) r\dr\,d\theta \end{multline*} \]
377 area of region $D\subset\R^2$ in the $xy$-plane. $T:D\ast\subset\R^2\rightarrow D$ and $T\left(u,v\right) =\left(x\left(u,v\right) ,y\left(u,v\right)\right)$ \[ \begin{align*} A\left(D\right) &= \iint_{D}\dx\dy\\ &= \iint_{D\ast} \abslr{ \frac{\partial\left(x,y\right)} {\partial\left(u,v\right)} }\du\dv \end{align*} \]
377
378
380 		Area of region in polar coordinates (fig. 6.2.4) \[ T:D\ast\rightarrow D\\ T\left(r,\theta\right)\\ =\left(r\cos{\theta},r\sin{\theta}\right)\\ D\ast=\left[0,1\right]\times\left[0,2\pi\right]\\ D=T\left(D\ast\right)\\ =\left\{\left(x,y\right):x^2+y^2\le1\right\} \] \[ A\left(D\right) =\iint_{D\ast}\left|\frac{\partial\left(x,y\right)}{\partial\left(u,v\right)}\right|\dr\,d\theta =\iint_{D\ast} r d r d\theta \]
385 Gaussian integral \[ \int_{-\infty}^{\infty}{e^{-x^2}dx}=\sqrt\pi \]
387 change of variables formula, three variables \[ \begin{multline*} \iint_{W}f\left(x,y,z\right)\dx\dy\dz\\ =\iint_{W\ast} f\left( x\left(u,v\right), y\left(u,v\right), z\left(u,v\right) \right)\\ \abs{ \frac{\partial\left(x,y,z\right)} {\partial\left(u,v,w\right)} } \du\dv\dw \end{multline*} \]
387 volume of region in $uvw-$space \[ \begin{multline*} V\left(W\right) =\iint_{W}\dx\dy\dz\\ =\iint_{W\ast} \abslr{ \frac{\partial\left(x,y,z\right)} {\partial\left(u,v,w\right)} }\du\dv\dw \end{multline*} \]
388 change of variables formula, cylindrical coordinates \[ \begin{multline*} \iint_{W} f\left(x,y,z\right)\dx\dy\dz\\ =\iint_{W\ast} f\left( r\cos{\theta}, r\sin{\theta}, z \right) r\dr\,d\theta\dz \end{multline*} \]
389 change of variables formula, spherical coordinates \[ \begin{multline*} \iint_{W}f\left(x,y,z\right) \dx\dy\dz\\ = \iint_{W\ast} f\left( \rho\sin{\varphi} \cos{\theta}, \rho\sin{\varphi}\sin{\theta}, \rho\cos{\varphi} \right)\\ \rho^2\sin{\varphi} \,d\rho \,d\theta \,d\varphi \end{multline*} \]

Notes

Page Notes
368 The change of variable formulas in this chapter are all based on the idea given in the first paragraph of the chapter. Read that paragraph well.
372 one-to-one map. $T:D\ast\rightarrow D$ with $u,v,u',v'\in D\ast$ \[ T\left(u,v\right)= T\left(u',v'\right) \Rightarrow u=u'\text{ and}v=v' \]
373 onto map. \[ D=T\left(D\ast\right) \]
371 theorem 1
382 theorem 2