Chapter 16: Vector CalculusContent check — Calculus (Guichard), Chapter 16: Vector Calculus

Compute ${\displaystyle {\int }_{C}x{y}^{2}ds}$ along the line segment from $(\mathrm{1,2,0})$ to $(\mathrm{2,1,3})$⁠.

Compute ${\displaystyle {\int }_C\sin xds}$ along the line segment from $(-\mathrm{1,2,1})$ to $(\mathrm{1,2,5})$⁠.

Compute ${\displaystyle {\int }_{C}z\cos (xy)ds}$ along the line segment from $(\mathrm{1,0,1})$ to $(\mathrm{2,2,3})$⁠.

Compute ${\displaystyle {\int }_C\sin xdx+\cos ydy}$ along the top half of the unit circle, from $(\mathrm{1,0})$ to $(-\mathrm{1,0})$⁠.

Compute ${\displaystyle {\int }_{C}x{e}^{y}dx+x^{2}ydy}$ along the line segment $y=3$⁠, $0\le x\le 2$⁠.

Compute ${\displaystyle {\int }_{C}x{e}^{y}dx+x^{2}ydy}$ along the line segment $x=4$⁠, $0\le y\le 4$⁠.

Compute ${\displaystyle {\int }_{C}x{e}^{y}dx+x^{2}ydy}$ along the curve $x=3t$⁠, $y=t^2$⁠, $0\le t\le 1$⁠.

Compute ${\displaystyle {\int }_{C}x{e}^{y}dx+x^{2}ydy}$ along the curve $⟨e^t,e^t⟩$⁠, $-1\le t\le 1$⁠.

Compute ${\displaystyle {\int }_C⟨\cos x,\sin y⟩\cdot d𝐫}$ along the curve $⟨t,t⟩$⁠, $0\le t\le 1$⁠.

Compute ${\displaystyle {\int }_C⟨1/xy,1/(x+y)⟩\cdot d𝐫}$ along the path from $(\mathrm{1,1})$ to $(\mathrm{3,1})$ to $(\mathrm{3,6})$ using straight line segments.

Compute ${\displaystyle {\int }_C⟨1/xy,1/(x+y)⟩\cdot d𝐫}$ along the curve $⟨2t,5t⟩$⁠, $1\le t\le 4$⁠.

Compute ${\displaystyle {\int }_C⟨1/xy,1/(x+y)⟩\cdot d𝐫}$ along the curve $⟨t,t^2⟩$⁠, $1\le t\le 4$⁠.

Compute ${\displaystyle {\int }_{C}yzdx+xzdy+xydz}$ along the curve $⟨t,t^2,t^3⟩$⁠, $0\le t\le 1$⁠.

An object moves from $(\mathrm{1,1})$ to $(\mathrm{4,8})$ along the path $𝐫(t)=⟨t^2,t^3⟩$⁠, subject to the force $𝐅=⟨x^2,\sin y⟩$⁠. Find the work done.

An object moves along the line segment from $(\mathrm{1,1})$ to $(\mathrm{2,5})$⁠, subject to the force $𝐅=$ $⟨x/(x^2+y^2),$ $y/(x^2+y^2)⟩$⁠. Find the work done.

An object moves along the parabola $𝐫(t)=⟨t,t^2⟩$⁠, $0\le t\le 1$⁠, subject to the force $𝐅=⟨1/(y+1),-1/(x+1)⟩$⁠. Find the work done.

An object moves along the line segment from $(\mathrm{0,0,0})$ to $(\mathrm{3,6,10})$⁠, subject to the force $𝐅=⟨x^2,y^2,z^2⟩$⁠. Find the work done.

An object moves along the curve $𝐫(t)=⟨\sqrt{t},1/\sqrt{t},t⟩$ $1\le t\le 4$⁠, subject to the force $𝐅=⟨y,z,x⟩$⁠. Find the work done.

An object moves from $(\mathrm{1,1,1})$ to $(\mathrm{2,4,8})$ along the path $𝐫(t)=⟨t,t^2,t^3⟩$⁠, subject to the force $𝐅=⟨\sin x,\sin y,\sin z⟩$⁠. Find the work done.

An object moves from $(\mathrm{1,0,0})$ to $(-\mathrm{1,0},\pi )$ along the path $𝐫(t)=⟨\cos t,\sin t,t⟩$⁠, subject to the force $𝐅=⟨y^2,y^2,xz⟩$⁠. Find the work done.

Find an $f$ so that $\nabla f=⟨2x+y^2,2y+x^2⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨x^3,-y^4⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨x{e}^y,y{e}^x⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨y\cos x,y\sin x⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨y\cos x,\sin x⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨x^2{y}^3,x{y}^4⟩$⁠, or explain why there is no such $f$⁠.

Find an $f$ so that $\nabla f=⟨yz,xz,xy⟩$⁠, or explain why there is no such $f$⁠.

Evaluate ${\displaystyle {\int }_C(10x^4-2x{y}^3)dx-3x^2{y}^{2}dy}$ where $C$ is the part of the curve $x^5-5x^2{y}^2-7x^2=0$ from $(3,-2)$ to $(\mathrm{3,2})$⁠.

Let $𝐅=⟨yz,xz,xy⟩$⁠. Find the work done by this force field on an object that moves from $(\mathrm{1,0,2})$ to $(\mathrm{1,2,3})$⁠.

Let $𝐅=$ $⟨e^y,$ $x{e}^y+$ $\sin z,$ $y\cos z⟩$⁠. Find the work done by this force field on an object that moves from $(\mathrm{0,0,0})$ to $(1,-\mathrm{1,3})$⁠.

Let $$𝐅=⟨\frac{-x}{{(x^2+y^2+z^2)}^{3/2}},\frac{-y}{{(x^2+y^2+z^2)}^{3/2}},\frac{-z}{{(x^2+y^2+z^2)}^{3/2}}⟩.$$ Find the work done by this force field on an object that moves from $(\mathrm{1,1,1})$ to $(\mathrm{4,5,6})$⁠.

Compute ${\displaystyle {\int }_{\partial D}2ydx+3xdy}$⁠, where $D$ is described by $0\le x\le 1$⁠, $0\le y\le 1$⁠.

Compute ${\displaystyle {\int }_{\partial D}xydx+xydy}$⁠, where $D$ is described by $0\le x\le 1$⁠, $0\le y\le 1$⁠.

Compute ${\displaystyle {\int }_{\partial D}{e}^{2x+3y}dx+e^{xy}dy}$⁠, where $D$ is described by $-2\le x\le 2$⁠, $-1\le y\le 1$⁠.

Compute ${\displaystyle {\int }_{\partial D}y\cos xdx+y\sin xdy}$⁠, where $D$ is described by $0\le x\le \pi /2$⁠, $1\le y\le 2$⁠.

Compute ${\displaystyle {\int }_{\partial D}{x}^{2}ydx+x{y}^{2}dy}$⁠, where $D$ is described by $0\le x\le 1$⁠, $0\le y\le x$⁠.

Compute ${\displaystyle {\int }_{\partial D}x\sqrt{y}dx+\sqrt{x+y}dy}$⁠, where $D$ is described by $1\le x\le 2$⁠, $2x\le y\le 4$⁠.

Compute ${\displaystyle {\int }_{\partial D}(x/y)dx+(2+3x)dy}$⁠, where $D$ is described by $1\le x\le 2$⁠, $1\le y\le x^2$⁠.

Compute ${\displaystyle {\int }_{\partial D}\sin ydx+\sin xdy}$⁠, where $D$ is described by $0\le x\le \pi /2$⁠, $x\le y\le \pi /2$⁠.

Compute ${\displaystyle {\int }_{\partial D}x\ln ydx}$⁠, where $D$ is described by $1\le x\le 2$⁠, ${\displaystyle e^x\le y\le e^{x^2}}$⁠.

Compute ${\displaystyle {\int }_{\partial D}\sqrt{1+x^2}dy}$⁠, where $D$ is described by $-1\le x\le 1$⁠, $x^2\le y\le 1$⁠.

Compute ${\displaystyle {\int }_{\partial D}{x}^{2}ydx-x{y}^{2}dy}$⁠, where $D$ is described by $x^2+y^2\le 1$⁠.

Compute ${\displaystyle {\int }_{\partial D}{y}^{3}dx+2x^{3}dy}$⁠, where $D$ is described by $x^2+y^2\le 4$⁠.

Evaluate ${\displaystyle {\oint }_C(y-\sin (x))dx+}$ ${\displaystyle \cos (x)dy}$⁠, where $C$ is the boundary of the triangle with vertices $(\mathrm{0,0})$⁠, $(\mathrm{1,0})$⁠, and $(\mathrm{1,2})$ oriented counter-clockwise.

Let $𝐅=⟨xy,-xy⟩$ and let $D$ be given by $0\le x\le 1$⁠, $0\le y\le 1$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Let $𝐅=⟨a{x}^2,b{y}^2⟩$ and let $D$ be given by $0\le x\le 1$⁠, $0\le y\le 1$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Let $𝐅=⟨a{y}^2,b{x}^2⟩$ and let $D$ be given by $0\le x\le 1$⁠, $0\le y\le x$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Let $𝐅=$ $⟨\sin x\cos y,$ $\cos x\sin y⟩$ and let $D$ be given by $0\le x\le \pi /2$⁠, $0\le y\le x$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Let $𝐅=⟨y,-x⟩$ and let $D$ be given by $x^2+y^2\le 1$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Let $𝐅=⟨x,y⟩$ and let $D$ be given by $x^2+y^2\le 1$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$ and ${\displaystyle {\int }_{\partial D}𝐅\cdot 𝐍ds}$⁠.

Find the area of the portion of $x+2y+4z=10$ in the first octant.

Find the area of the portion of $2x+4y+z=0$ inside $x^2+y^2=1$⁠.

Find the area of $z=x^2+y^2$ that lies below $z=1$⁠.

Find the area of $z=\sqrt{x^2+y^2}$ that lies below $z=2$⁠.

Find the area of the portion of $x^2+y^2+z^2=a^2$ that lies in the first octant.

Find the area of the portion of $x^2+y^2+z^2=a^2$ that lies above $x^2+y^2\le b^2$⁠, $b\le a$⁠.

Find the area of $z=x^2-y^2$ that lies inside $x^2+y^2=a^2$⁠.

Find the area of $z=xy$ that lies inside $x^2+y^2=a^2$⁠.

Find the area of $x^2+y^2+z^2=a^2$ that lies above the interior of the circle given in polar coordinates by $r=a\cos \theta$⁠.

Find the area of the cone $z=k\sqrt{x^2+y^2}$ that lies above the interior of the circle given in polar coordinates by $r=a\cos \theta$⁠.

Find the area of the plane $z=ax+by+c$ that lies over a region $D$ with area $A$⁠.

Find the area of the cone $z=k\sqrt{x^2+y^2}$ that lies over a region $D$ with area $A$⁠.

Find the area of the cylinder $x^2+z^2=a^2$ that lies inside the cylinder $x^2+y^2=a^2$⁠.

Find the center of mass of an object that occupies the upper hemisphere of $x^2+y^2+z^2=1$ and has density $x^2+y^2$⁠.

Find the center of mass of an object that occupies the surface $z=xy$⁠, $0\le x\le 1$⁠, $0\le y\le 1$ and has density $\sqrt{1+x^2+y^2}$⁠.

Find the center of mass of an object that occupies the surface ${\displaystyle z=\sqrt{x^2+y^2}}$⁠, $1\le z\le 4$ and has density $x^{2}z$⁠.

Find the centroid of the surface of a right circular cone of height $h$ and base radius $r$⁠, not including the base.

Evaluate ⁠, where $D$ is given by $z=x^2+y^2$⁠, $-1\le x\le 1$⁠, $-1\le y\le 1$⁠, oriented up.

Evaluate ⁠, where $D$ is given by $z=3x-5y$⁠, $1\le x\le 2$⁠, $0\le y\le 2$⁠, oriented up.

Evaluate ⁠, where $D$ is given by $z=1-x^2-y^2$⁠, $x^2+y^2\le 1$⁠, oriented up.

Evaluate ⁠, where $D$ is given by $z=x+y^2+2$⁠, $0\le x\le 1$⁠, $x\le y\le 1$⁠, oriented up.

Evaluate ⁠, where $D$ is given by $z=xy$⁠, $0\le x\le 1$⁠, $-x\le y\le x$⁠, oriented up.

Evaluate ⁠, where $D$ is given by $z=a^2-x^2-y^2$⁠, $x^2+y^2\le b^2$⁠, oriented up.

A fluid has density 870 kg/m⁠${}^3$ and flows with velocity $𝐯=⟨z,y^2,x^2⟩$⁠, where distances are in meters and the components of $𝐯$ are in meters per second. Find the rate of flow outward through the portion of the cylinder $x^2+y^2=4$⁠, $0\le z\le 1$ for which $y\ge 0$⁠.

Gauss's Law says that the net charge, $Q$⁠, enclosed by a closed surface, $S$⁠, is where $𝐄$ is an electric field and ${ϵ}_0$ (the permittivity of free space) is a known constant; $𝐍$ is oriented outward. Use Gauss's Law to find the charge contained in the cube with vertices $(\pm 1,\pm 1,\pm 1)$ if the electric field is $𝐄=⟨x,y,z⟩$⁠.

Let $𝐅=⟨z,x,y⟩$⁠. The plane $z=2x+2y-1$ and the paraboloid $z=x^2+y^2$ intersect in a closed curve. Stokes's Theorem implies that where the line integral is computed over the intersection $C$ of the plane and the paraboloid, and the two surface integrals are computed over the portions of the two surfaces that have boundary $C$ (provided, of course, that the orientations all match). Compute all three integrals.

Let $D$ be the portion of $z=1-x^2-y^2$ above the $x$⁠-⁠$y$ plane, oriented up, and let $𝐅=⟨x{y}^2,-x^{2}y,xyz⟩$⁠. Compute ⁠.

Let $D$ be the portion of $z=2x+5y$ inside $x^2+y^2=1$⁠, oriented up, and let $𝐅=⟨y,z,-x⟩$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$⁠.

Compute ${\displaystyle {\oint }_C{x}^{2}zdx+3xdy-y^{3}dz}$⁠, where $C$ is the unit circle ${\displaystyle x^2+y^2=1}$ oriented counter-clockwise.

Let $D$ be the portion of $z=px+qy+r$ over a region in the $x$⁠-⁠$y$ plane that has area $A$⁠, oriented up, and let $𝐅=$ $⟨ax+$ $by+$ $cz,$ $ax+$ $by+$ $cz,$ $ax+$ $by+$ $cz⟩$⁠. Compute ${\displaystyle {\int }_{\partial D}𝐅\cdot d𝐫}$⁠.

Using ${\displaystyle 𝐅=⟨3x,y^3,-2z^2⟩}$ and the region bounded by ${\displaystyle x^2+y^2=9}$⁠, $z=0$⁠, and $z=5$⁠, compute both integrals from the Divergence Theorem.

Let $E$ be the volume described by $0\le x\le a$⁠, $0\le y\le b$⁠, $0\le z\le c$⁠, and $𝐅=⟨x^2,y^2,z^2⟩$⁠. Compute ⁠.

Let $E$ be the volume described by $0\le x\le 1$⁠, $0\le y\le 1$⁠, $0\le z\le 1$⁠, and $𝐅=⟨2xy,3xy,z{e}^{x+y}⟩$⁠. Compute ⁠.

Let $E$ be the volume described by $0\le x\le 1$⁠, $0\le y\le x$⁠, $0\le z\le x+y$⁠, and $𝐅=⟨x,2y,3z⟩$⁠. Compute ⁠.

Let $E$ be the volume described by $x^2+y^2+z^2\le 4$⁠, and $𝐅=⟨x^3,y^3,z^3⟩$⁠. Compute ⁠.

Let $E$ be the hemisphere described by $0\le z\le \sqrt{1-x^2-y^2}$⁠, and $𝐅=$ $⟨\sqrt{x^2+y^2+z^2},$ $\sqrt{x^2+y^2+z^2},$ $\sqrt{x^2+y^2+z^2}⟩$⁠. Compute ⁠.

Let $E$ be the volume described by $x^2+y^2\le 1$⁠, $0\le z\le 4$⁠, and $𝐅=⟨x{y}^2,yz,x^{2}z⟩$⁠. Compute ⁠.

Let $E$ be the solid cone above the $x$⁠-⁠$y$ plane and inside $z=1-\sqrt{x^2+y^2}$⁠, and $𝐅=$ $⟨x{\cos }^{2}z,$ $y{\sin }^{2}z,$ $\sqrt{x^2+y^2}z⟩$⁠. Compute ⁠.