Chapter 9

a) Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) be a continuous function in the xy-plane. Find an integral expression for \(\iint_{\mathrm{G}(\mathrm{R})} \mathrm{f}(\mathrm{x}, \mathrm{y}) \mathrm{dxdy}\) in polar coordinates using the change of variables formula. b) Repeat for \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})\) and \(\iiint \mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) \mathrm{dxdydz}\) represented by spherical coordinates.

Let \(\mathrm{S}\) be the volume defined by \(\mathrm{S}=\left\\{\mathrm{x}^{2}+\mathrm{y}^{2} \leq 1,0 \leq \mathrm{z} \leq 2\right\\}\). Find the integral \(\iiint_{\mathrm{S}}\left(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\right) \mathrm{d} \mathrm{x} \mathrm{dy} \mathrm{d} \mathrm{z}\).

Let \(\mathrm{x}=\mathrm{e}^{\mathrm{u}} \cos \mathrm{v}\) \(\mathrm{y}=\mathrm{e}^{\mathrm{u}} \sin \mathrm{v}\) Find \(\iint_{R^{*}} \mathrm{x}^{2} \mathrm{dx}\) dy where \(\mathrm{R}^{*}\) is the image of the rectangle \(\mathrm{R}\) in uv-space consisting of \(\\{(\mathrm{u}, \mathrm{v}) \mid 0 \leq \mathrm{u} \leq 1\) and \(0 \leq \mathrm{v} \leq \pi\\}\).

Find \(\iint_{D^{*}} \exp [(x-y) /(x+y)] d x\) dy where \(D^{*}\) is the region bounded by the lines \(\mathrm{x}=0, \mathrm{y}=0\), and \(\mathrm{x}+\mathrm{y}=1\).

Consider the transformation \(\mathrm{x}-\mathrm{u}+\mathrm{v} \quad \mathrm{y}=\mathrm{v}-\mathrm{u}^{2}\) Let \(D\) be the set in the \(u-v\) plane bounded by the lines \(\mathrm{u}=0, \mathrm{v}=0\), and \(\mathrm{u}+\mathrm{v}=2\) Find the area of \(\mathrm{D}^{*}\), the image of \(\mathrm{D}\), directly and by a change of variables.

Compute the area of the paraboloid given by the equation \(z=x^{2}+y^{2}\), with \(0 \leq z \leq 2\)

Find the area of the upper hemisphere of the sphere given by the equation \(x^{2}+y^{2}+z^{2}=3^{2}\).

Compute the total area of the torus given parametrically by \(\mathrm{x}=(\mathrm{a}+\mathrm{b} \cos \varphi) \cos \theta ; \mathrm{y}=(\mathrm{a}+\mathrm{b} \cos \varphi) \sin \theta ; \mathrm{z}=\mathrm{b} \sin \varphi\) \(0

Find the area of the torus whose parametrization is given by \(\mathrm{x}=(\mathrm{R}-\cos \mathrm{v}) \cos \mathrm{u} \quad-\pi \leq \mathrm{u} \leq \pi\) \(\mathrm{y}=(\mathrm{R}-\cos \mathrm{v}) \sin \mathrm{u}\) \(-\pi \leq \mathrm{v} \leq \pi\) \(z=\sin v\) where \(\mathrm{R}>1\).

Let \(\mathrm{S}\) be the surface defined by \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\) and let the unit normal vector function have representations directed away from the origin. Compute the integral of the function \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{a}\) over \(\mathrm{S} .\)

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 \(\int_{\mathrm{C}} \mathrm{F}^{-} \cdot \mathrm{ds}^{-}=0\).

Verify Stokes's Theorem for \(\mathrm{F}^{-}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left(3 \mathrm{y},-\mathrm{xz}, \mathrm{yz}^{2}\right)\) where \(\mathrm{S}\) is the surface \(2 \mathrm{z}=\mathrm{x}^{2}+\mathrm{y}^{2}\) bounded by \(\mathrm{z}=2\) and \(\mathrm{C}\) is its boundary.

Let \(\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left\\{-\mathrm{y} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\},\left\\{\mathrm{x} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\}, 0\right]\) Evaluate \(\oint_{C} F^{-} \cdot d r^{-}\) where \(C\) is the circle \(x^{2}+y^{2}=1 .\) Also evaluate \(\int_{\mathrm{S}}\left(\operatorname{curl} \mathrm{F}^{-}\right) \cdot \mathrm{n}^{\rightarrow} \mathrm{d} \mathrm{A}\) and explain the results.

Prove: \(\oint_{C} F^{\rightarrow} \cdot d r^{\rightarrow}=0\) for every closed curve \(C\) if and only if \(\operatorname{curl} \mathrm{F}^{\rightarrow}=0\).

Show that the 2 -form $$ \sigma=\left[(x d y d z+y d z d x+z d x d y) /\left\\{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}\right\\}\right] $$ satisfies \(\mathrm{d} \sigma=0\) but that \(\sigma\) is not exact. Do this by proving that \(\iint_{\mathrm{S}} \sigma\), where \(\mathrm{S}\) is the unit sphere, is not zero.