Chapter 9

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G} $$

In Problems, evaluate \(\int_{c} \mathbf{F} \cdot d \mathbf{r}\). \(\mathbf{F}(x, y, z)=(y-y z \sin x) \mathbf{i}+(x+z \cos x) \mathbf{j}+y \cos x \mathbf{k}\) \(\mathbf{r}(t)=2 t \mathbf{i}+(1+\cos t)^{2} \mathbf{j}+4 \sin ^{3} t \mathbf{k}, 0 \leq t \leq \pi / 2\)

Evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve \(C\) between \((0,0,0)\) and \((6,8,5)\). \(C\) consists of the line segments from \((0,0,0)\) to \((2,3,4)\) and from \((2,3,4)\) to \((6,8,5)\).

Find the points on the given surtace at which the tangent plane is parallel to the indicated plane. $$ x^{2}+y^{2}+z^{2}=7 ; 2 x+4 y+6 z=1 $$

Find the first partial derivatives of the given function. $$ f(x, y)=\frac{3 x-y}{x+2 y} $$

In Problems, find parametric equations of the tangent line to the given curve at the indicated value of \(t\). $$ x=t, y=\frac{1}{2} t^{2}, z=\frac{1}{3} t^{3} ; t=2 $$

In Problems, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x^{2}+4 x z+2 y z^{2} ;(1,2,-1) $$

Evaluate the given double integral by means of an appropriate change of variables. \(\iint_{R}(6 x+3 y) d A\), where \(R\) is the trapezoidal region in the first quadrant with vertices \((1,0),(4,0),(2,4)\), and \(\left(\frac{1}{2}, 1\right)\)

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-3}^{3} \int_{0}^{\sqrt{9-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). $$ \begin{aligned} &\mathbf{F}(x, y, z)=(y-y z \sin x) \mathbf{i}+(x+z \cos x) \mathbf{j}+y \cos x \mathbf{k} \\ &\mathbf{r}(t)=2 t \mathbf{i}+(1+\cos t)^{2} \mathbf{j}+4 \sin ^{3} t \mathbf{k}, 0 \leq t \leq \pi / 2 \end{aligned} $$

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G} $$

Find the points on the given surface at which the tangent plane is parallel to the indicated plane. $$ x^{2}+y^{2}+z^{2}=7 ; 2 x+4 y+6 z=1 $$

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x^{2}+4 x z+2 y z^{2} ;(1,2,-1) $$

Discuss the curvature near a point of inflection of \(y=F(x)\).

Find parametric equations of the tangent line to the given curve at the indicated value of \(t\). $$ x=t, y=\frac{1}{2} t^{2}, z=\frac{1}{3} t^{3} ; t=2 $$

In Problems 23-26, evaluate the given double integral by means of an appropriate change of variables. \(\begin{aligned} &\iint_{R}(x+y)^{4} e^{x-y} d A, \text { where } R \text { is the square region with vertices }\\\ &(1,0),(0,1),(1,2), \text { and }(2,1) \end{aligned}\)

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{\sqrt{2} / 2} \int_{0}^{\sqrt{1-y^{2}}} \frac{y^{2}}{\sqrt{x^{2}+y^{2}}} d x d y $$

Find the volume of the solid bounded by the graphs of the given equations. $$ z=4-y^{2}, x=3, x=0, y=0, z=0, \text { first octant } $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \times(\mathbf{F}+\mathbf{G})=\nabla \times \mathbf{F}+\nabla \times \mathbf{G} $$

Evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve \(C\) between \((0,0,0)\) and \((6,8,5)\). $$ x=3 t, y=t^{3}, z=\frac{5}{4} t^{2}, \quad 0 \leq t \leq 2 $$

Find the points on the given surtace at which the tangent plane is parallel to the indicated plane. $$ x^{2}-2 y^{2}-3 z^{2}=33 ; 8 x+4 y+6 z=5 $$

Find the first partial derivatives of the given function. $$ f(x, y)=\frac{x y}{\left(x^{2}-y^{2}\right)^{2}} $$

In Problems, find parametric equations of the tangent line to the given curve at the indicated value of \(t\). $$ x=t^{3}-t, y=\frac{6 t}{t+1}, z=(2 t+1)^{2} ; t=1 $$

In Problems, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x y z ;(3,1,-5) $$

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{\sqrt{2 / 2}} \int_{0}^{\sqrt{1-y^{2}}} \frac{y^{2}}{\sqrt{x^{2}+y^{2}}} d x d y $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). $$ \begin{aligned} &\mathbf{F}(x, y, z)=\left(2-e^{z}\right) \mathbf{i}+(2 y-1) \mathbf{j}+\left(2-x e^{2}\right) \mathbf{k} \\ &\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k},(-1,1,-1) \text { to }(2,4,8) \end{aligned} $$

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \times(\mathbf{F}+\mathbf{G})=\nabla \times \mathbf{F}+\nabla \times \mathbf{G} $$

Find the points on the given surface at which the tangent plane is parallel to the indicated plane. $$ x^{2}-2 y^{2}-3 z^{2}=33 ; 8 x+4 y+6 z=5 $$

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x y z ;(3,1,-5) $$

Find parametric equations of the tangent line to the given curve at the indicated value of \(t\). $$ x=t^{3}-t, y=\frac{6 t}{t+1}, z=(2 t+1)^{2} ; t=1 $$

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} e^{x^{2}+y^{2}} d x d y $$

Use Green's theorem to evaluate the given double integral by means of a line integral. [Hint: Find appropriate functions \(P\) and \(Q\).] $$ \begin{aligned} &\iint_{R} x^{2} d A ; R \text { is the region bounded by the ellipse }\\\ &x^{2} / 9+y^{2} / 4 \quad 1 \end{aligned} $$

Find the mass of the given surface with the indicated density function. \(S\) that portion of the plane \(x+y+z=1\) in the first octant; density at a point \(P\) directly proportional to the square of the distance from the \(y z\) -plane

Find the volume of the solid bounded by the graphs of the given equations. $$ x^{2}+y^{2}=4, x-y+2 z=4, x=0, y=0, z=0, \text { first octant } $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \cdot(f \mathbf{F})=f(\nabla \cdot \mathbf{F})+\mathbf{F} \cdot \nabla f $$

The inverse square law of gravitational attraction between two masses \(m_{1}\) and \(m_{2}\) is given by \(\mathbf{F}=-G m_{1} m_{2} \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Show that \(\mathbf{F}\) is conservative. Find a potential function for \(\mathbf{F}\).

The velocity of a particle moving in a fluid is described by means of a velocity field \(\mathbf{v}=v_{1} \mathbf{i}+v_{2} \mathbf{j}+v_{3} \mathbf{k}\), where the components \(v_{1}, v_{2}\), and \(v_{3}\) are functions of \(x, y, z\), and time \(t\). If the velocity of the particle is \(\mathbf{v}(t)=6 t^{2} x \mathbf{i}-4 t y^{2} \mathbf{j}+2 t(z+1) \mathbf{k}\) find \(\mathbf{r}(t) .\)

Find the first partial derivatives of the given function. $$ g(u, v)=\ln \left(4 u^{2}+5 v^{3}\right) $$

In Problems, find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right] $$

In Problems, find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ f(x, y)=\tan \left(x^{2}+y^{2}\right) ;(\sqrt{\pi / 6}, \sqrt{\pi / 6}) $$

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{1}-y^{2}} e^{x^{2}+y^{2}} d x d y $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). The inverse square law of gravitational attraction between two masses \(m_{1}\) and \(m_{2}\) is given by \(\mathbf{F}=-G m_{1} m_{2} \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Show that \(\mathbf{F}\) is conservative. Find a potential function for \(\mathbf{F}\).

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \cdot(f \mathbf{F})=f(\nabla \cdot \mathbf{F})+\mathbf{F} \cdot \nabla f $$

Find points on the surface \(x^{2}+4 x+y^{2}+z^{2}-2 z=11\) at which the tangent plane is horizontal.

Find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ f(x, y)=\tan \left(x^{2}+y^{2}\right) ;(\sqrt{\pi / 6}, \sqrt{\pi / 6}) $$

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right] $$

Use \(V=\iiint_{D} d V\) andthe substitutions \(u=x / a, v=y / b, w=z / c\) to show that the volume of theellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} l c^{2}=1\) is \(V=\frac{4}{3} \pi a b c\).

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-\sqrt{\pi}}^{\sqrt{\pi}} \int_{0}^{\sqrt{\pi-x^{2}}} \sin \left(x^{2}+y^{2}\right) d y d x $$

Use Green's theorem to evaluate the given double integral by means of a line integral. [Hint: Find appropriate functions \(P\) and \(Q\).] \(\iint_{R}[1-2(y-1)] d A ; R\) is the region in the first quadrant bounded by the circle \(x^{2}+(y-1)^{2} \quad 1\) and \(x \quad 0\)

Find the mass of the given surface with the indicated density function. \(S\) the hemisphere \(z=\sqrt{4-x^{2}-y^{2}} ; \rho(x, y, z)=|x y|\)