91Ó°ÊÓ

Find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\). $$ u=x y, v=y, w=x+z $$

Evaluate the triple integral. \(\iiint_{G} x y z d V,\) where \(G\) is the solid in the first octant that \(G\) is bounded by the parabolic cylinder \(z=2-x^{2}\) and the planes \(z=0, y=x,\) and \(y=0\)

Find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\). $$ u=x+y+z, v=x+y-z, w=x-y+z $$

Evaluate the triple integral. \(\iiint \cos (z / y) d V,\) where \(G\) is the solid defined by the inequalities \(\pi / 6 \leq y \leq \pi / 2, y \leq x \leq \pi / 2,0 \leq z \leq x y\)

Evaluate the iterated integrals. $$ \int_{3}^{4} \int_{1}^{2} \frac{1}{(x+y)^{2}} d y d x $$

Sketch the parametric surface. $$ \begin{array}{l}{\text { (a) } x=u, y=v, z=u^{2}+v^{2}} \\ {\text { (b) } x=u, y=u^{2}+v^{2}, z=v} \\ {\text { (c) } x=u^{2}+v^{2}, y=u, z=v}\end{array} $$

Find a parametric representation of the surface in terms of the parameters \(u=x\) and \(v=y .\) $$ \text { (a) } 2 z-3 x+4 y=5 \quad \text { (b) } z=x^{2} $$

Use spherical coordinates to find the volume of the solid. The solid bounded above by the sphere \(\rho=4\) and below by the cone \(\phi=\pi / 3\).

Evaluate the double integral over the rectangular region R. $$ \iint_{R} 4 x y^{3} d A ; R=\\{(x, y):-1 \leq x \leq 1,-2 \leq y \leq 2\\} $$

Determine whether the statement is true or false. Explain your answer. If \(\mathbf{r}=x(u, v) \mathbf{i}+y(u, v) \mathbf{j},\) then evaluating \(|\partial(x, y) / \partial(u, v)|\) at a point \(\left(u_{0}, v_{0}\right)\) gives the perimeter of the parallelogram generated by the vectors \(\partial \mathbf{r} / \partial u\) and \(\partial \mathbf{r} / \partial v\) at \(\left(u_{0}, v_{0}\right)\).