Chapter 9: Chapter 9

Find the integral of the function $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}\( over the surface \)z=x^{2}+y\( with \)x, y$ satisfying the inequalities \(0 \leq x \leq 1\) and \(-1 \leq y \leq 1\)

Let \(\mathrm{S}\) be the hemisphere given by $\mathrm{S}:\left\\{(\mathrm{x}, \mathrm{y}, \mathrm{z}) \mid \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\right.$ \(=1, z>0\\}\). Let \(f\) be the function \(f(x, y, z)=x^{2} y^{2} z\). Compute the integral \(\iint_{\mathrm{S}} \mathrm{f} \mathrm{d} \mathrm{A}\).

Integrate the function \(z\) over the surface \(z=x^{2}+y^{2}\) with $x^{2}+y^{2} \leq 1$

Compute the integral of the vector field $\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( \)=\left(\mathrm{y},-\mathrm{x}, \mathrm{z}^{2}\right)$ over the paraboloid \(\mathrm{z}=\mathrm{x}^{2}+\mathrm{y}^{2}\) with \(0 \leq \mathrm{z} \leq 1\)

Let the unit hemisphere be parametrized by $$ \begin{array}{ll} x=\cos u \sin v & 0

Let \(\mathrm{S}\) be the surface \(\mathrm{S}=\left[\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\right\\}\) and let \(\mathrm{f}\) be the function $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\(. Let \)\nabla \mathrm{f}^{-}$ represent the gradient of \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) .\) Compute the integral $\iint_{\mathrm{S}} \nabla \mathrm{f} \cdot \mathrm{n}^{-} \mathrm{d} \mathrm{A}$.

Given that \(\mathrm{S}\) is the surface of a region \(\mathrm{U}\) for which the divergence theorem is applicable, let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( be any point of \)S$ and let 0 be any fixed point in space. Show that the volume of \(\mathrm{U}\) is given by $\mathrm{V}=(1 / 3) \iint_{\mathrm{S}} \mathrm{r} \cos \varphi \mathrm{d} \mathrm{A}$, where \(\varphi\) is the angle between the directed line \(\mathrm{OP}\) and the outer normal S at \(\mathrm{P}\), and \(\mathrm{r}\) is the distance \(\mathrm{OP}\).

Let \(U\) be the interior of a closed surface \(S\). Let \(f, g\) be functions. Let \(\nabla \mathrm{f}\) be the gradient of \(\mathrm{f}\) and $\nabla^{2} \mathrm{f}\( be the divergence of the gradient of \)\mathrm{f}$. Prove: a) $\iint_{\mathrm{S}} \mathrm{f}(\nabla \mathrm{g}) \cdot \mathrm{n}^{-} \mathrm{da}=\iiint_{\mathrm{U}}\left(\mathrm{f} \nabla^{2} \mathrm{~g}+\nabla \mathrm{f} \cdot \nabla \mathrm{g}\right) \mathrm{dV}$. b) $\iint_{\mathrm{S}}(\mathrm{f} \nabla \mathrm{g}-\mathrm{g} \nabla \mathrm{f}) \cdot \mathrm{n}^{-} \mathrm{da}=\iiint_{\mathrm{U}}\left(\mathrm{f} \nabla^{2} \mathrm{~g}-\mathrm{g} \nabla^{2} \mathrm{f}\right) \mathrm{dV}$.

Verify Stokes's Theorem for the vector field $\mathrm{F}^{-}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( \)=(\mathrm{z}, \mathrm{x}, \mathrm{y})$ where \(\mathrm{S}\) is defined by $\mathrm{z}=4-\mathrm{x}^{2}-\mathrm{y}^{3}, \mathrm{z} \geq 0$

Let $\mathrm{F}^{-}=\left(\mathrm{F}_{1}, \mathrm{~F}_{2}, \mathrm{~F}_{3}\right)$ be a vector field that satisfies the following conditions $$ \begin{gathered} \left(\partial \mathrm{F}_{2} / \partial \mathrm{z}\right)=\left(\partial \mathrm{F}_{3} / \partial \mathrm{y}\right) ;\left(\partial \mathrm{F}_{3} / \partial \mathrm{x}\right)=\left(\partial \mathrm{F}_{1} / \partial \mathrm{z}\right) ;\left(\partial \mathrm{F}_{2} / \partial \mathrm{x}\right) \\\ =\left(\partial \mathrm{F}_{1} / \partial \mathrm{y}\right), \end{gathered} $$ on a region bounded by a curve c. (See Fig. 1). Prove, using Stokes's Theorem that \(\int_{\mathrm{C}} \mathrm{F}^{-} \cdot \mathrm{ds}^{-}=0\).