Question

Describe the vector field \( \mathbf{F} (x,y)=3\mathbf{i+j}\) by drawing some of its vectors.

Question

Describe the vector field \(\mathbf{F} (x,y)=\sin x\mathbf{i+j}\) by drawing some of its vectors.

Question

Find \(\int_{C}\left( \dfrac{dx}{y}+\dfrac{dy}{x}\right)\), where \(C\) is the arc of the parabola \(y=x^{2}\) from \((1,1)\) to \((2,4)\).

Question

Find \(\int_{C}\left[ y\cos (xy) \,dx+x\cos (xy) \,dy\right]\), where \(C\) is the curve \(x=t^{2}\), \( y=t^{3}\), \(0\leq t\leq 1\).

Question

1. Confirm that the line integral \(\int_{C}\left[ y\cos (xy) \,dx+ x\cos (xy) \,dy\right] \) is independent of the path.
2. Find \(\int_{C}\left[ y\cos (xy) \,dx+x\cos (xy) \,dy\right] \) by following the right-angle path from \((0,0)\) to \( (1,0)\) to \((1,1)\).

Question

1. Find a function \(f\) whose gradient is \(y\cos (xy) \,\mathbf{i}+x\cos (xy) \,\mathbf{j}\).
2. Explain why \(\int_{C}[y\cos (xy) \,dx+x\cos (xy) \,dy]=f(1,1)-f(0,0)\), where \(C\) is a smooth curve joining (\(0,0\)) to (\(1,1\)).

Question

Find \(\int_{C}[(yz-y-z)\,dx+(xz-x-z)\,dy+(xy-x-y)\,dz]\), where \(C\) is the twisted cubic \(x=t\), \(y=t^{2}\), \(z=t^{3}\), \(0\leq t\leq 1\).

Question

Find \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}\) if \(\ \mathbf{F}(x,y,z)=xy\,\mathbf{i}+xz\,\mathbf{j}+(y-x)\,\mathbf{k}\) and \(C\) is the line segment from \(\left( 0,0,0\right) \) to \(\left( 1,-2,3\right) \).

Question

1. Show that \(f(x,y) =3x^{2}+y-2\) is a potential function of \(\mathbf{F}(x,y) =6x\mathbf{i}+\,\mathbf{ j}\).
2. Use the Fundamental Theorem of Line Integrals to find \( \int_{C}[6xdx+\,dy]\), where \(C\) is any curve joining the two points \(( 4,1) \) to \(( 6,3)\).

Question

Find \(\int_{C}[e^{x}\sin y\,dx+e^{x}\cos y\,dy]\), where \(C\) is the arc of the parabola \(y=x^{2}\) from \((0,0)\) to \((1,1)\).
(Hint: \( \boldsymbol{\nabla \,}( e^{x}\sin y) = e^{x}\sin y\mathbf{i}+e^{x}\cos y\mathbf{j.})\)

Question

Mass of a Wire Find the mass of a thin piece of wire in the shape of a circular arc \(x=\cos t\), \(y=\sin t\), \(0\leq t\leq \dfrac{3\pi }{4} \), if the variable mass density of the wire is \(\rho (x,y)=x+1\).

Question

Find the line integral \(\int_{C}( x^{3}dx+y^{3}dy)\)

1. if \(C\) consists of the line segment from \((1,1)\) to \((2,4)\).
2. if \(C\) consists of line segments from \((1,1)\) to \((1,5),\) from \( (1,5) \) to \(\left( 2,3\right) \) and from \(\left( 2,3\right) \) to \((2,4) \).
3. if \(C\) is a part of the parabola \(x=t\), \(y=t^{2}\), \(\ 1\leq t\leq 2.\)

Question

Mass of a Lamina  A lamina is in the shape of the cone \(z=\sqrt{ x^{2}+y^{2}},\) \(1\leq z\leq 4\). If the mass density of the lamina is \(\rho (x,y,z)=x^{2}+y^{2}\), find the mass \(M\) of the lamina.

Question

Find \(\int_{C}[x^{2}y^{3}\,dx-xy^{4}\,dy]\), where \(C\) is the arc of the parabola \(y^{2}=x\) from \((0,0)\) to \((1,1)\).

Question

Find \(\int_{C}x^{2}y^{2}\,ds; C: x=\cos t, y=\sin t; 0\leq t\leq \pi.\)

Question

Work If \(\mathbf{F}(x,y)=\dfrac{y}{(x^{2}+y^{2})^{3/2}}\mathbf{ i}-\dfrac{x}{(x^{2}+y^{2})^{3/2}}\mathbf{j}\), find the work done by going around the unit circle against \(\mathbf{F}\) from \(t=0\) to \(t=2\pi \).

Question

Work Find the work done by the force \(\mathbf{F}=y\sin x \mathbf{i}+\sin x\mathbf{j}\) in moving an object along the curve \(y=\sin x\) from \(x=0\) to \(x=2\pi\).

Question

Work Find the work done by the force \[ \mathbf{F}=\dfrac{x}{x^{2}+y^{2}}\mathbf{i}+\dfrac{y}{x^{2}+y^{2}}\mathbf{j} \] in moving an object along the curve \(\mathbf{r}(t)=t\cos t\mathbf{i}+t\sin t\mathbf{j}\) from \((-\pi ,0)\) to \((2\pi ,0)\).

Question

Area Use Green's Theorem to find the area of the region enclosed by the curves \(C_{1}\): \(x(t) =t,\) \(y(t) =t^{2}+3\) and \(C_{2}\): \(x(t) =t,\) \(y(t) =30-2t^{2}.\)

Question

Use Green's Theorem to find the line integral \(\int_{C}\left[ \ln \left( 1+y\right) dx+\dfrac{xy}{1+y}dy\right] \) where \(C\) is the parallelogram with vertices \( (0,0) \), \((2,1) ,\) \(( 2,6) ,\) and \( ( 0,5)\) traversed counterclockwise.

Question

Find \(\oint_{C}(y^{2}\,dx-x^{2}\,dy)\), where \(C\) is the square with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) traversed counterclockwise. Do not use Green's Theorem.

Question

Rework Problem 21 using Green's Theorem.

Question

Find \(\oint_{C}[(x-y)\,dx+(x+y)\,dy]\), where \(C\) is the ellipse \(x=2\cos t\), \(y=3\sin t\), \(0\leq t\leq 2\pi \), without using Green's Theorem.

Question

Rework Problem 23 using Green's Theorem.

Question

Area Use Green's Theorem to find the area of the multiply connected region enclosed by the ellipse \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{16} =1\) which has a small circular hole given by \(x^{2}+\left( y-1\right) ^{2}=1\) punched out of its interior.

Question

1. Identify the coordinate curves of the surface parametrized by \(\mathbf{r}(u,v) =3u\cos v\mathbf{i} +2u\sin v\mathbf{j}+u^{2}\mathbf{k}\), \(\ 0\leq u\leq 1,\) \(0\leq v\leq 2\pi \).
2. Find a rectangular equation for the surface.

Question

Find a parametrization of the part of the cylinder \( 4x^{2}+25y^{2}=100\) that lies above the plane \(z=1\) and below the plane \(z=6\).

Question

1. Find an equation of the tangent plane to the surface \( \mathbf{r}(u,v) =u\sin v\,\mathbf{i}+u^{2}\,\mathbf{j}+u\cos v\, \mathbf{k},\) at the point \((\sqrt{3},4,1) \).
2. Find an equation of the normal line to the tangent plane at the point \((\sqrt{3},4,1)\).

Question

Find the surface area of the part of the paraboloid \(\mathbf{r} (u,v) =u\sin v\,\mathbf{i}+u^{2}\,\mathbf{j}+u\cos v\,\mathbf{k,} \) \(0\leq u\leq 4,\) \(0\leq v\leq 2\pi\).

Question

Find \(\iint\limits_{\kern-3ptS}x\,dS,\) where \(S\) is the surface parametrized by \(\mathbf{r}(u,v)=\cos v\,\mathbf{i}+3\sin u\sin v\,\mathbf{j} +3\cos u\sin v\mathbf{k}\), \(0\leq u\leq 2\pi \) and \(0\leq v\leq \pi /2\).

Question

Find \(\iint\limits_{\kern-3ptS}z^{2}\,dS\), where \(S\) is the sphere \(x^{2}+y^{2}+z^{2}=4\).

Question

Find the outer unit normal vectors to the surface \(S\) that forms the boundary of the solid \(z=f(x,y)=\sqrt{25-x^{2}-y^{2}},\) \( 0\leq x^{2}+y^{2}\leq 25.\)

Question

Find \(\iint\limits_{\kern-3ptS}x\,dS\), where \(S\) is the surface \( x^{2}+y^{2}=9\), \(-1\leq z\leq 1\).

Question

Find \(\iint\limits_{\kern-3ptS}z\,dS\), where \(S\) is the surface \( z=9-x-y\), \(x^{2}+y^{2}\leq 9\).

Question

Find \(\iint\limits_{\kern-3ptS}\cos x\,dS\), where \(S\) is the portion of the plane \(x=y+z,\) \(x\leq \pi \), \(\ y\geq 0,\,\ z\geq 0\).

Question

A fluid has a constant mass density \(\rho .\) Find the mass of fluid flowing across the surface \(x^{2}+y^{2}=1,\) \(\ 0\leq z\leq 1\) in a unit of time, in the direction outward from the \(z\)-axis if the velocity of the fluid at any point on the surface is \(\mathbf{F}=x^{2}\mathbf{i}+y \mathbf{j}-z\mathbf{k}\).

Question

\(\mathbf{F}=z\cos x\mathbf{i}+\sin y\mathbf{j}+e^{x}\mathbf{k}\)

Question

\(\mathbf{F}=x^{2}\mathbf{i}-3y\mathbf{j}+4z^{2}\mathbf{k}\)

Question

\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\)

Question

\(\mathbf{F}=xe^{y}\mathbf{i}-ye^{z}\mathbf{j}+ze^{x}\mathbf{k}\)

Question

Use the Divergence Theorem to find \(\iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS\), where \(S \) is the surface bounded by \(x^{2}+y^{2}=1\) and \(0\leq z\leq 1,\) and \(\mathbf{F}=x^{2} \mathbf{i}+y\,\mathbf{j}-z\mathbf{k}\).

Question

1. Find a function \(f(x,y,z)\) whose gradient is \((yz-y-z) \mathbf{i}+(xz-x-z)\mathbf{j}+(xy-x-y)\mathbf{k}\).
2. Use Stokes' Theorem to confirm the answer to \( \int_{C}[(yz-y-z)\,dx+(xz-x-z)\,dy+(xy-x-y)\,dz]\), where \(C\) is the twisted cubic \(x=t\), \(y=t^{2}\), \(z=t^{3}\), \(0\leq t\leq 1\).

Question

Find \(\iint\limits_{\kern-3ptS}(xz\cos \alpha +yz\cos \beta +x^{2}\cos \gamma )\,dS\), where \(S\) is the upper half of the unit sphere together with the plane \(z=0,\) and \(\cos \alpha ,\) \(\cos \beta ,\) and \(\cos \gamma \) are the direction cosines for the outer unit normal to \(S\).

Question

Find \(\iiint\limits_{\kern-13ptE}{div}\mathbf{F}\,dV\), where \(E\) is the unit ball, \(\left\Vert \mathbf{r}\right\Vert \leq 1\), and \(\mathbf{F}= {\Vert \mathbf{r\Vert }}^{2}\mathbf{r}\).

Question

Use Stokes' Theorem to find \(\int_{C}[(x-y)\,dx+(y-z)\,dy+ (z-x) \,dz]\), where \(C\) is the boundary of the portion of the plane \(x+y+z=1,\) \( x\geq 0,\) \(y\geq 0,\) \(z\geq 0\) (traversed counterclockwise when viewed from above).

Question

Let \(\mathbf{F}(x,y,z)=x^{3}\mathbf{i}+y^{3}\mathbf{j}+z^{3} \mathbf{k}\) be the velocity of a fluid flow in\br space, where the mass density of the fluid is \(1\).

1. Find the flux across the sphere \(x^{2}+y^{2}+z^{2}=1\).
2. Find the circulation around the circle \(x^{2}+y^{2}=1\) in the \( xy\)-plane.

Question

Determine if \(\mathbf{F}( x,y,z) =yz\mathbf{i}+xy \mathbf{j}+xy\mathbf{k}\) is a conservative vector field.

Question

Let \(\mathbf{F}( x,y,z) =2xy^{2}z\,\mathbf{i} +2x^{2}yz\,\mathbf{j}+(x^{2}y^{2}-2z)\mathbf{k}\). Show that \(\mathbf{F}\) is a conservative vector field.