一般换元与雅可比矩阵

The general substitution principle

A worked linear change

Cylindrical and spherical in space

Choosing a substitution

For $x = \tfrac{u+v}{2}$, $y = \tfrac{u-v}{2}$, what is $|J|$?

Under $u = x + y$, $v = x - y$, the diamond $|x| + |y| \le 1$ maps to which region?

Spherical coordinates carry which volume element?

Using spherical coordinates, $\iiint \rho^2 \sin\varphi\, d\rho\, d\varphi\, d\theta$ over the ball of radius $2$ gives $c \cdot \pi$. Give $c$, as a fraction like 32/3.

The change of variables turns $\iint_{|x|+|y| \le 1} (x + y)^2\, dA$ into $\tfrac{1}{2}\int_{-1}^1\int_{-1}^1 u^2\, du\, dv$. Give the value, as a fraction like 2/3.