Chapter 9

In Problems, evaluate the given integral. $$ \int \frac{1}{1+t^{2}}\left(\mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k}\right) d t $$

Verify that the given function satisfies the wave equation: $$a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}$$ $$ u=\cos (x+a t)+\sin (x-a t) $$

Consider the gravitational potential $$ U(x, y)=\frac{-G m}{\sqrt{x^{2}+y^{2}}} $$ where \(G\) and \(m\) are constants. Show that \(U\) increases or decreases most rapidly along a line through the origin.

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(2, \frac{5 \pi}{6},-3\right) $$

Find parametric equations for the normal line at the indicated point. In Problems 35 and 36, find symmetric equations for the normal line. $$ x^{2}+y^{2}-z^{2}=0 ;(3,4,5) $$

Evaluate the given integral. $$ \int \frac{1}{1+t^{2}}\left(\mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k}\right) d t $$

In Problems \(35-38\), convert the point given in cylindrical coardinates to rectangular cocrdinates. $$ \left(\sqrt{3}, \frac{\pi}{3},-4\right) $$

Let \(T(x, y, z)=x^{2}+y^{2}+z^{2}\) represent temperature and let the "flow" of heat be given by the vector field \(\mathbf{F}=-\nabla T\). Find the flux of heat out of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\). [Hint: The surface area of a sphere of radius \(a\) is \(\left.4 \pi a^{2} .\right]\)

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{2} \int_{y^{2}}^{4} \cos \sqrt{x^{3}} d x d y $$

Any scalar function \(f\) for which \(\nabla^{2} f=0\) is said to be harmonic. Verify that \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\) is harmonic except at the origin. \(\nabla^{2} f=0\) is called Laplace's equation.

Verify that the line integral \(\int_{C} y^{2} d x+x y d y\) has the same value on \(C\) for each of the following parameterizations: $$ \begin{array}{lll} C: x=2 t+1, & y=4 t+2, & 0 \leq t \leq 1 \\ C: x=t^{2}, & y=2 t^{2}, & 1 \leq t \leq \sqrt{3} \\ C: x=\ln t, & y=2 \ln t, & e \leq t \leq e^{3}. \end{array} $$

In Problems, find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime}(t)=6 \mathbf{i}+6 t \mathbf{j}+3 t^{2} \mathbf{k} ; \mathbf{r}(0)=\mathbf{i}-2 \mathbf{j}+\mathbf{k} $$

Show that every normal line to the graph \(x^{2}+y^{2}+z^{2}=a^{2}\) passes through the origin.

Verify that the given function satisfies the wave equation: $$a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}$$ The molecular concentration \(C(x, t)\) of a liquid is given by \(C(x, t)=t^{-1 / 2} e^{-x^{2} / k t} .\) Verify that this function satisfies the diffusion equation: $$ \frac{k}{4} \frac{\partial^{2} C}{\partial x^{2}}=\frac{\partial C}{\partial t} $$

If \(f(x, y)=x^{3}-12 x+y^{2}-10 y\), find all points at which \(\|\nabla f\|=0 .\)

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(\sqrt{3}, \frac{\pi}{3},-4\right) $$

The molecular concentration \(C(x, t)\) of a liquid is given by \(C(x, t)=t^{-1 / 2} e^{-x^{2} / k t}\). Verify that this function satisfies the diffusion equation: $$ \frac{k}{4} \frac{\partial^{2} C}{\partial x^{2}}=\frac{\partial C}{\partial t} $$

Find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime}(t)=6 \mathbf{i}+6 t \mathbf{j}+3 t^{2} \mathbf{k} ; \mathbf{r}(0)=\mathbf{i}-2 \mathbf{j}+\mathbf{k} $$

In Problems \(35-38\), convert the point given in cylindrical coardinates to rectangular cocrdinates. $$ \left(4, \frac{7 \pi}{4}, 0\right) $$

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} x \sqrt{1-x^{2}-y^{2}} d y d x $$

Verify that $$f(x, y)=\arctan \left(\frac{2}{x^{2} y^{2}-1}\right), \quad x^{2}+y^{2} \neq 1$$ satisfies Laplace's equation in two variables $$ \nabla^{2} f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0. $$

In Problems, find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime}(t)=t \sin t^{2} \mathbf{i}-\cos 2 t \mathbf{j} ; \mathbf{r}(0)=\frac{3}{2} \mathbf{i} $$

Two surfaces are said to be orthogonal at a point \(P\) of intersection if their normal lines at \(P\) are orthogonal. Prove that the surfaces given by \(F(x, y, z)=0\) and \(G(x, y, z)=0\) are orthogonal at \(P\) if and only if \(F_{x} G_{x}+F_{y} G_{y}+F_{z} G_{z}=0\)

The pressure \(P\) exerted by an enclosed ideal gas is given by \(P=k(T / V)\), where \(k\) is a constant, \(T\) is temperature, and \(V\) is volume. Find: (a) the rate of change of \(P\) with respect to \(V\), (b) the rate of change of \(V\) with respect to \(T\), and (c) the rate of change of \(T\) with respect to \(P\).

Suppose $$ \begin{gathered} D_{\mathrm{u}} f(a, b)=7, \quad D_{\mathrm{v}} f(a, b)=3 \\ \mathbf{u}=\frac{5}{13} \mathbf{i}-\frac{12}{13} \mathbf{j}, \quad \mathbf{v}=\frac{5}{13} \mathbf{i}+\frac{12}{13} \mathbf{j} \end{gathered} $$ Find \(\nabla f(a, b)\).

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(4, \frac{7 \pi}{4}, 0\right) $$

Consider the three curves between \((0,0)\) and \((2,4)\) : $$ \begin{array}{lll} C_{1}: x=t, & y=2 t, & 0 \leq t \leq 2 \\ C_{2}: x=t, & y=t^{2}, & 0 \leq t \leq 2 \\ C_{3}: x=2 t-4, & y=4 t-8, & 2 \leq t \leq 3 \end{array} $$ Show that \(\int_{C_{1}} x y d s=\int_{C_{3}} x y d s\), but \(\int_{C_{1}} x y d s \neq \int_{C_{2}} x y d s\).

Suppose $$ \begin{gathered} D_{\mathbf{u}} f(a, b)=7, \quad D_{\mathbf{v}} f(a, b)=3 \\ \mathbf{u}=\frac{5}{13} \mathbf{i}-\frac{12}{13} \mathbf{j}, \quad \mathbf{v}=\frac{5}{13} \mathbf{i}+\frac{12}{13} \mathbf{j} . \end{gathered} $$ Find \(\nabla f(a, b)\)

Find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime}(t)=t \sin t^{2} \mathbf{i}-\cos 2 t \mathbf{j} ; \mathbf{r}(0)=\frac{3}{2} \mathbf{i} $$

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (1,-1,-9) $$

Coulomb's law states that the electric field \(\mathbf{E}\) due to a point charge \(q\) at the origin is given by \(\mathbf{E}=k q \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(k\) is a constant and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Determine the flux out of a sphere \(x^{2}+y^{2}+z^{2}=a^{2}\)

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{1} \int_{x}^{1} \frac{1}{1+y^{4}} d y d x $$

Let \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) be the position vector of a mass \(m_{1}\) and let the mass \(m_{2}\) be located at the origin. If the force of gravitational attraction is $$\mathbf{F}=-\frac{G m_{1} m_{2}}{\|\mathbf{r}\|^{3}} \mathbf{r}$$ verify that \(\operatorname{curl} \mathbf{F}=\mathbf{0}\) and \(\operatorname{div} \mathbf{F}=0, \mathbf{r} \neq \mathbf{0}\)

Assume a smooth curve \(C\) is described by the vector function \(\mathbf{r}(t)\) for \(a \leq t \leq b .\) Let acceleration, velocity, and speed be given by \(\mathbf{a}=d \mathbf{v} / d t, \mathbf{v}=d \mathbf{r} / d t\), and \(v=\|\mathbf{v}\|\), respectively. Using Newton's second law \(\mathbf{F}=m \mathbf{a}\), show that, in the absence of friction, the work done by \(\mathbf{F}\) in moving a particle of constant mass \(m\) from point \(A\) at \(t=a\) to point \(B\) at \(t=b\) is the same as the change in kinetic energy: $$ K(B)-K(A)=\frac{1}{2} m[v(b)]^{2}-\frac{1}{2} m[v(a)]^{2}. $$

In Problems, find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime \prime}(t)=12 t \mathbf{i}-3 t^{-1 / 2} \mathbf{j}+2 \mathbf{k} ; \mathbf{r}^{\prime}(1)=\mathbf{j}, \mathbf{r}(1)=2 \mathbf{i}-\mathbf{k} $$

Use the Chain Rule to find the indicated partial derivatives. $$ z=e^{u v^{2}} ; u=x^{3}, v=x-y^{2} ; \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} $$

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (1,-1,-9) $$

Find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \mathbf{r}^{\prime \prime}(t)=12 t \mathbf{i}-3 t^{-1 / 2} \mathbf{j}+2 \mathbf{k} ; \mathbf{r}^{\prime}(1)=\mathbf{j}, \mathbf{r}(1)=2 \mathbf{i}-\mathbf{k} $$

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (2 \sqrt{3}, 2,17) $$

If \(\sigma(x, y, z)\) is charge density in an electrostatic field, then the total charge on a surface \(S\) is \(Q=\iint_{S} \sigma(x, y, z) d S\). Find the total charge on that part of the hemisphere \(z=\sqrt{16-x^{2}-y^{2}}\) that is inside the cylinder \(x^{2}+y^{2}=9\) if the charge density at a point \(P\) on the surface is directly proportional to distance from the \(x y\) -plane.

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{x^{3}+1} d x d y $$

In Problems, find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \begin{aligned} &\mathbf{r}^{\prime \prime}(t)=\sec ^{2} t \mathbf{i}+\cos t \mathbf{j}-\sin t \mathbf{k} ; \\ &\mathbf{r}^{\prime}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{r}(0)=-\mathbf{j}+5 \mathbf{k} \end{aligned} $$

Use the Chain Rule to find the indicated partial derivatives. $$ z=u^{2} \cos 4 v ; u=x^{2} y^{3}, v=x^{3}+y^{3} ; \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} $$

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (2 \sqrt{3}, 2,17) $$

If \(\rho(x, y)\) is the density of a wire (mass per unit length), then \(m=\int_{C} \rho(x, y) d s\) is the mass of the wire. Find the mass of a wire having the shape of the semicircle \(x=1+\cos t, y=\sin t\), \(0 \leq t \leq \pi\), if the density at a point \(P\) is directly proportional to distance from the \(y\)-axis.

Find a vector function \(\mathbf{r}\) that satisfies the indicated conditions. $$ \begin{aligned} &\mathbf{r}^{\prime \prime}(t)=\sec ^{2} t \mathbf{i}+\cos t \mathbf{j}-\sin t \mathbf{k} \\ &\mathbf{r}^{\prime}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{r}(0)=-\mathbf{j}+5 \mathbf{k} \end{aligned} $$

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (-\sqrt{2}, \sqrt{6}, 2) $$

Find the center of mass of the lamina that has the given shape and density. $$ x=0, x=4, y=0, y=3 ; \rho(x, y)=x y $$

In Problems, find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}+\text { ct } \mathbf{k} ; 0 \leq t \leq 2 \pi $$

Use the Chain Rule to find the indicated partial derivatives. $$ z=4 x-5 y^{2} ; \quad x=u^{4}-8 v^{3}, y=(2 u-v)^{2} ; \quad \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v} $$