# User:Melikamp/ma225

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## Parametric Equations

1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).

x=7t and y=17t, where 0 ≤ t ≤ 1

2. Find parametric equations describing the line segment from $P(x_1,y_1)$ to $Q(x_2,y_2)$.

$x=x_1 + (x_2-x_1)t \mbox{ and } y=y_1 + (y_2-y_1)t, \mbox{ where } 0 \leq t \leq 1$

3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.

$x=3\cos(-t),\ y=1.5\sin(-t)$

## Polar Coordinates

20. Convert the equation into Cartesian coordinates: $r=\sin(\theta)\sec^2(\theta).$

$y=x^2$

21. Find an equation of the line y=mx+b in polar coordinates.

$r=\frac{b}{\sin(\theta)-m\cos(\theta)}$

Sketch the following polar curves without using a computer.

22. $r=2-2\sin(\theta)$
23. $r^2 = 4\cos(\theta)$
24. $r=2\sin(5\theta)$

Sketch the following sets of points.

25. $\{(r,\theta):\theta=2\pi/3\}$
26. $\{(r,\theta):|\theta|\leq\pi/3\mbox{ and }|r|<3\}$

## Calculus in Polar Coordinates

Find points where the following curves have vertical or horizontal tangents.

40. $r=4\cos(\theta)$

Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)

41. $r=2+2\sin(\theta)$

Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/4)

Sketch the region and find its area.

42. The region inside the limaçon $2+\cos(\theta)$
9π/2
43. The region inside the petals of the rose $4\cos(2\theta)$ and outside the circle $r=2$
$8\pi/3 + 4\sqrt3$

## Vectors and Dot Product

60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)

$(x-1)^2+(y-2)^2+z^2 = 33$

61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
62. Find the value of $|\mathbf u+3\mathbf v|$ if $\mathbf u = \langle 1,3,0\rangle$ and $\mathbf v = \langle 3,0,2\rangle$

$\sqrt{145}$

63. Find all unit vectors parallel to $\langle 1,2,3\rangle$

$\pm\frac{1}{\sqrt{14}}\langle 1,2,3\rangle$

64. Prove one of the distributive properties for vectors in $\mathbb R^3$: $c(\mathbf u + \mathbf v) = c\mathbf u+c\mathbf v$

$eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray$

65. Find all unit vectors orthogonal to $3\mathbf i+4\mathbf j$ in $\mathbb R^2$

$\pm\left\langle \frac{-4}5, \frac35\right\rangle$

66. Find all unit vectors orthogonal to $3\mathbf i+4\mathbf j$ in $\mathbb R^3$

$\left\langle \frac45c, -\frac35c, \sqrt{1-c^2}\right\rangle,\ c\in[-1,1]$

67. Find all unit vectors that make an angle of $\pi/3$ with the vector $\langle 1,2\rangle$

$\frac{\sqrt5}{10}\left\langle 1 \pm 2\sqrt3,\ 2 \mp \sqrt3 \right\rangle$

## Cross Product

Find $\mathbf u\times\mathbf v$ and $\mathbf v\times\mathbf u$

80. $\mathbf u = \langle -4, 1, 1\rangle$ and $\mathbf v = \langle 0,1,-1\rangle$

$\mathbf u \times \mathbf v = \langle -2,-4,-4\rangle$

81. $\mathbf u = \langle 1,2,-1\rangle$ and $\mathbf v = \langle 3,-4,6\rangle$

$\mathbf u = \langle 8,-9,-10\rangle$

Find the area of the parallelogram with sides $\mathbf u$ and $\mathbf v$.

82. $\mathbf u = \langle -3, 0, 2\rangle$ and $\mathbf v = \langle 1,1,1\rangle$

$\sqrt{38}$

83. $\mathbf u = \langle 8, 2, -3\rangle$ and $\mathbf v = \langle 2,4,-4\rangle$

$6\sqrt{41}$

84. Find all vectors that satisfy the equation $\langle 1,1,1\rangle\times\mathbf u = \langle 0,1,1\rangle$

None

85. Find the volume of the parallelepiped with edges given by position vectors $\langle 5,0,0\rangle$, $\langle 1,4,0\rangle$, and $\langle 2,2,7\rangle$

$140$

86. A wrench has a pivot at the origin and extends along the x-axis. Find the magnitude and the direction of the torque at the pivot when the force $\mathbf F = \langle 1,2,3\rangle$ is applied to the wrench n units away from the origin.

$\mathbf\tau = \langle 0,-3n,2n\rangle$, so the torque is directed along $\pm\langle 0,-3,2\rangle$

Prove the following identities or show them false by giving a counterexample.

87. $\mathbf u\times (\mathbf u\times \mathbf v) = \mathbf 0$

False: $\mathbf i\times(\mathbf i\times\mathbf j) = -\mathbf j$

88. $\mathbf u\cdot (\mathbf v \times \mathbf w) = \mathbf w\cdot (\mathbf u \times \mathbf v)$

Once expressed in component form, both sides evaluate to $u_1v_2w_3 - u_1v_3w_2 + u_2v_3w_1 - u_2v_1w_3 + u_3v_1w_2 - u_3v_2w_1$

89. $(\mathbf u - \mathbf v)\times(\mathbf u + \mathbf v) = 2(\mathbf u\times\mathbf v)$

$eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray$

## Calculus of Vector-Valued Functions

100. Differentiate $\mathbf r(t) = \langle te^{-t}, t\ln t, t\cos(t)\rangle$.

$\langle e^{-t}-te^{-t}, \ln(t) + 1,cos(t)-t\sin(t)\rangle$

101. Find a tangent vector for the curve $\mathbf r(t) = \langle 2t^4, 6t^{3/2}, 10/t \rangle$ at the point $t = 1$.

$\langle 8, 9, -10\rangle$

102. Find the unit tangent vector for the curve $\mathbf r(t) = \langle t, 2, 2/t \rangle,\ t \neq 0$.

$\displaystyle\frac{\langle t^2,0,-2\rangle}{\sqrt{t^4+4}}$

103. Find the unit tangent vector for the curve $\mathbf r(t) = \langle \sin(t), \cos(t), e^{-t}\rangle,\ t\in[0,\pi]$ at the point $t = 0$.

$\displaystyle\frac{\langle 1,0,-1\rangle}{\sqrt 2}$

104. Find $\mathbf r$ if $\mathbf r'(t) = \langle \sqrt t, \cos(\pi t), 4/t\rangle$ and $\mathbf r(1) = \langle 2,3,4\rangle$.

$\displaystyle\left\langle \frac{2t^{3/2}+4}{3}, \frac{\sin(\pi t)}{\pi}+3, 4\ln|t|+4\right\rangle$

105. Evaluate $\displaystyle\int_0^{\ln 2}(e^{-t}\mathbf i+2e^{2t}\mathbf j-4e^{t}\mathbf k)dt$

$\langle 1/2, 3, -4\rangle$

## Motion in Space

120. Find velocity, speed, and acceleration of an object if the position is given by $\mathbf r(t) = \langle 3\sin(t),5\cos(t),4\sin(t)\rangle$.

$\mathbf v = \langle 3\cos(t), -5\sin(t), 4\cos(t)\rangle$, $|\mathbf v| = 5$, $\mathbf a = \langle -3\sin(t), -5\cos(t), -4\sin(t)\rangle$

121. Find the velocity and the position vectors for $t\geq 0$ if the acceleration is given by $\mathbf a(t) = \langle e^{-t},1\rangle,\ \mathbf v(0) = \langle 1,0\rangle,\ \mathbf r(0) = \langle 0,0\rangle$.

$\mathbf v(t) = \langle 2-e^{-t}, t\rangle$, $\mathbf r(t) = \langle e^{-t}+2t-1, t^2/2\rangle$

## Length of Curves

Find the length of the following curves.

140. $\mathbf r(t) = \langle4\cos(3t),4\sin(3t)\rangle,\ t \in [0,2\pi/3].$

$8\pi$

141. $\mathbf r(t) = \langle2+3t,1-4t,3t-4\rangle,\ t \in [1,6].$

$5\sqrt{34}$

## Parametrization and Normal Vectors

142. Find a description of the curve that uses arc length as a parameter: $\mathbf r(t) = \langle t^2,2t^2,4t^2\rangle\ t\in[1,4].$

$\displaystyle\mathbf r(s) = \left(\frac s{\sqrt{21}}+1\right)\langle 1,2,4\rangle$

143. Find the unit tangent vector T and the principal unit normal vector N for the curve $\mathbf r(t) = \langle t^2,t\rangle.$ Check that TN=0.

$\mathbf T(t) = \displaystyle\frac{\langle 2t, 1\rangle}{\sqrt{4t^2+1}},\ \mathbf N(t) = \displaystyle\frac{\langle 1, -2t\rangle}{\sqrt{4t^2+1}}$

## Equations of Lines And Planes

160. Find an equation of a plane passing through points $(1,1,2),\ (1,2,2),\ (-1,0,1).$

$x-2z+3=0$

161. Find an equation of a plane parallel to the plane 2xy+z=1 passing through the point (0,2,-2)

$2x-y+z+4=0$

162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).

$\mathbf r(t) = \langle 5+t,5+t,5+2t\rangle$

163. Find an equation of the line where planes x+2yz=1 and x+y+z=1 intersect.

$\mathbf r(t) = \langle 1-3t,2t,t \rangle$

164. Find the angle between the planes x+2yz=1 and x+y+z=1.

$\cos^{-1}{\frac 2{\sqrt{18}}}$

165. Find the distance from the point (3,4,5) to the plane x+y+z=1.

$\frac{11}3\sqrt 3$

## Limits And Continuity

Evaluate the following limits.

180. $\displaystyle\lim_{(x,y)\rightarrow(1,-2)}\frac{y^2+2xy}{y+2x}$

−2

181. $\displaystyle\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}$

1/6

At what points is the function f continuous?

182. $f(x,y) = \ln|x-y|$

$\{(x,y)\mid x\neq y\}$

183. $f(x,y) = \displaystyle\frac{\ln(x^2 + y^2)}{x-y+1}$

All points (x,y) except for (0,0) and the line y=x+1

Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)

184. $\displaystyle\lim_{(x,y)\rightarrow(0,0)}\frac{4xy}{3x^2+y^2}$

The limit is 1 along the line y=x, and −1 along the line y=−x

185. $\displaystyle\lim_{(x,y)\rightarrow(0,0)}\frac{y}{\sqrt{x^2-y^2}}$

The limit is 0 along the line y=0, and $1/\sqrt 3$ along the line x=2y

186. $\displaystyle\lim_{(x,y)\rightarrow(0,0)}\frac{x^3-y^2}{x^3+y^2}$

The limit is 1 along the line y=0, and −1 along the line x=0

187. $\displaystyle\lim_{(x,y)\rightarrow(0,0)}\frac{x^2y^2+y^6}{x^3}$

The limit is 0 along any line of the form y=mx, and 2 along the parabola $x=y^2$

## Partial Derivatives

200. Find $\partial z / \partial x$ if $\displaystyle z(x,y) = \frac1{\ln(xy)}$

$\frac{-1}{x(\ln(xy))^2}$

201. Find all three partial derivatives of the function $\displaystyle f(x,y,z) = xe^{y^2 + z}$

$\displaystyle f_x=e^{y^2+z},\ f_y=2xye^{y^2+z},\ f_z=f.$

Find the four second partial derivatives of the following functions.

202. $f(x,y) = \cos(xy)$

$f_{xx}=-y^2\cos(xy),\ f_{yy} = -x^2\cos(xy),\ f_{xy} = f_{yx} = -\sin(xy) -xy\cos(xy).$

203. $f(x,y) = xe^y$

$f_{xx}=0,\ f_{yy} = xe^y,\ f_{xy} = f_{yx} = e^y.$

## Chain Rule

Find $df/dt.$

220. $f(x,y) = x^2y-xy^3,\ x(t) = t^2,\ y(t) = t^{-2}$

$2t+4t^{-5}$

221. $f(x,y) = \sqrt{x^2+y^2},\ x(t) = \cos(2t),\ y(t) = \sin(2t)$

0

222. $\displaystyle f(x,y,z) = \frac{x-y}{y+z},\ x(t) = t,\ \displaystyle y(t) = 2t,\ z(t) = 3t$

0

Find $f_s,\ f_t.$

223. $f(x,y) = \sin(x)\cos(2y),\ x=s+t,\ y=s-t$

Failed to parse (syntax error): f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)

224. $\displaystyle f(x,y,z) = \frac{x-z}{y+z},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t$

$-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2$

225. The volume of a pyramid with a square base is $V = \frac13x^2h$, where x is the side of the square base and h is the height of the pyramid. Suppose that $\displaystyle x(t) = \frac t{t+1}$ and $\displaystyle h(t) = \frac1{t+1}$ for $t\geq 0.$ Find $V'(t).$

$\displaystyle \frac{2t-t^2}{3(t+1)^4}$

## Tangent Planes

Find an equation of a plane tangent to the given surface at the given point(s).

240. $xy\sin(z) = 1,\ (1,2,\pi/6),\ (-1,-2,5\pi/6).$

$(x-1)+\frac12(y-2)+\sqrt3(z-\pi/6) = 0,\ (x+1)+\frac12(y+2)+\sqrt3(z-5\pi/6) = 0$

241. $z = x^2e^{x-y},\ (2,2,4),\ (-1,-1,1).$

$-8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0$

242. $z = \tan^{-1}(x+y),\ (0,0,0).$

$x+y-z=0$

243. $\sin(xyz) = 1/2,\ (\pi,1,1/6).$

$x+\pi y+6\pi z=3\pi$

## Maximum And Minimum Problems

Find critical points of the function f. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

260. $f(x,y) = x^4 + 2y^2 - 4xy$

Local minima at (1,1) and (−1,−1), saddle at (0,0)

261. $f(x,y) = \tan^{-1}(xy)$

262. $f(x,y) = 2xye^{-x^2-y^2}$

Saddle at (0,0), local maxima at $(\pm1/\sqrt2,\pm1/\sqrt2),$ local minima at $(\pm1/\sqrt2,\mp1/\sqrt2)$

Find absolute maximum and minimum values of the function f on the set R.

263. $f(x,y) = x^2+y^2-2y+1,\ R=\{(x,y)\mid x^2+y^2\leq 4\}$

Maximum of 9 at (0,−2) and minimum of 0 at (0,1)

264. $f(x,y) = x^2+y^2-2x-2y,$ R is a closed triangle with vertices (0,0), (2,0), and (0,2).

Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)

265. Find the point on the plane xy+z=2 closest to the point (1,1,1).

$(4/3,2/3,4/3)$

266. Find the point on the surface $z = x^2+y^2+10$ closest to the plane $x+2y-z=0.$

$(0.5, 1, 11.25)$

## Double Integrals over Rectangular Regions

Evaluate the given integral over the region R.

280. $\displaystyle\iint_R (x^2+xy)dA,\ R = \{(x,y)\mid x\in[1,2],\ y\in[-1,1]\}$

$14/3$

281. $\displaystyle\iint_R (xy\sin(x^2))dA,\ R = \{(x,y)\mid x\in[0,\sqrt{\pi/2}],\ y\in[0,1]\}$

$1/4$

282. $\displaystyle\iint_R \frac{x}{(1+xy)^2}dA,\ R = \{(x,y)\mid x\in[0,4],\ y\in[1,2]\}$

$\ln(5/3)$

Evaluate the given iterated integrals.

283. $\displaystyle\int_0^2\int_0^1 x^5y^2e^{x^3y^3}dydx$

$(e^8-9)/9$

284. $\displaystyle\int_1^4\int_0^2 e^{y\sqrt x}dydx$

$e^4-e^2-2$

## Double Integrals over General Regions

Evaluate the following integrals.

300. $\displaystyle\iint_R xy dA,$ R is bounded by x=0, y=2x+1, and y=5−2x.

$2$

301. $\displaystyle\iint_R (x+y) dA,$ R is in the first quadrant and bounded by x=0, $y=x^2,$ and $y=8 - x^2.$

$152/3$

Use double integrals to compute the volume of the given region.

302. The solid in the first octant bound by the coordinate planes and the surface $z = 8-x^2-2y^2.$

$4\pi\sqrt2$

303. The solid beneath the cylinder $z=y^2$ and above the region $R = \{(x,y)\mid y\in[0,1],\ x\in[y,1]\}.$

$1/12$

304. The solid bounded by the paraboloids $z=x^2+y^2$ and $z = 50-x^2-y^2.$

$625\pi$

## Double Integrals in Polar Coordinates

320. Evaluate $\displaystyle\iint_R 2xy dA$ for $R=\{(r,\theta)\mid r\in[1,3],\ \theta\in[0,\pi/2]\}$

$20$

321. Find the average value of the function $f(r,\theta) = 1/r^2$ over the region $\{(r,\theta)\mid r\in[2,4]\}.$

$\displaystyle\frac{\ln2}6$

322. Evaluate $\displaystyle\int_0^3\int_0^{\sqrt{9-x^2}} \sqrt{x^2+y^2}dydx.$

$9\pi/2$

323. Evaluate $\displaystyle\iint_R \frac{x-y}{x^2+y^2+1}dA$ if R is the unit disk centered at the origin.

$0$

## Triple Integrals

340. Evaluate $\displaystyle\int_1^{\ln 8}\int_0^{\ln 4}\int_0^{\ln 2} e^{-x-y-2z}dxdydz.$

$\displaystyle\frac3{16}\left(e^{-2}-8^{-2}\right)$

In the following exercises, sketching the region of integration may be helpful.

341. Find the volume of the solid in the first octant bounded by the plane 2x+3y+6z=12 and the coordinate planes.

$8$

342. Find the volume of the solid in the first octant bounded by the cylinder $z=\sin(y)$ for $y\in[0,\pi]$, and the planes y=x and x=0.

$\pi$

343. Evaluate $\displaystyle\int_0^1\int_y^{2-y}\int_0^{2-x-y} xydzdxdy.$

$2/15$

344. Rewrite the integral $\displaystyle\int_0^1\int_{-2}^2\int_0^{\sqrt{4-y^2}}dzdydx$ in the order dydzdx.

$\displaystyle\int_0^1\int_0^2\int_{-\sqrt{4-z^2}}^{\sqrt{4-z^2}}dydzdx$

## Cylindrical And Spherical Coordinates

360. Evaluate the integral in cylindrical coordinates: $\displaystyle\int_0^3\int_{0}^{\sqrt{9-x^2}}\int_0^{\sqrt{x^2+y^2}}\frac1{\sqrt{x^2+y^2}}dzdydx$

$9\pi/4$

361. Find the mass of the solid cylinder $D = \{(r,\theta,z)\mid r\in[0,3],\ z\in[0,2]\}$ given the density function $\delta(r,\theta,z) = 5e^{-r^2}$

$10\pi(1-e^{-9})$

362. Use a triple integral to find the volume of the region bounded by the plane z=0 and the hyperboloid $z = \sqrt{17} - \sqrt{1+x^2+y^2}$

$\displaystyle\frac{2\pi(1+7\sqrt{17})}{3}$

363. If D is a unit ball, use a triple integral in spherical coordinates to evaluate $\iiint_D(x^2+y^2+z^2)^{5/2}dV$

$\pi/2$

364. Find the mass of a solid cone $\{(\rho,\phi,\theta)\mid \phi\leq\pi/3,\ z\in[0,4]\}$ if the density function is $\delta(\rho,\phi,\theta) = 5-z$

$128\pi$

365. Find the volume of the region common to two cylinders: $x^2+z^2 = 1,\ y^2+z^2 = 1$

$16/3$

## Center of Mass and Centroid

380. Find the center of mass for three particles located in space at (1,2,3), (0,0,1), and (1,1,0), with masses 2, 1, and 1 respectively.

$\frac{\langle 3,5,7\rangle}{4}$

381. Find the center of mass for a piece of wire with the density $\rho(x) = 1+\sin(x)$ for $x\in[0,\pi].$

$\pi/2$

382. Find the center of mass for a piece of wire with the density $\rho(x) = 2-x^2/16$ for $x\in[0,4].$

$9/5$

383. Find the centroid of the region in the first quadrant bounded by the coordinate axes and $x^2+y^2=16.$

$\left(\frac{16}{3\pi},\frac{16}{3\pi}\right)$

384. Find the centroid of the region in the first quadrant bounded by $y=\ln(x)$, $y=0$, and $x=e$.

$((e^2+1)/4, e/2 - 1)$

385. Find the center of mass for the region $\{(x,y)\mid x\in[0,4], y\in[0,2]\}$, with the density $\rho(x,y) = 1+x/2.$

$(7/3, 1)$

386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density $\rho(x,y) = 1+x+y.$

$(16/11, 16/11)$

## Vector Fields

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

401. Find and sketch the gradient field $\mathbf F = \nabla\phi$ for the potential function $\phi(x,y) = \sqrt{x^2+y^2}$.

$\mathbf F = \left\langle\frac x{\sqrt{x^2+y^2}},\frac y{\sqrt{x^2+y^2}}\right\rangle$

402. Find and sketch the gradient field $\mathbf F = \nabla\phi$ for the potential function $\phi(x,y) = \sin(x)\sin(y)$ for $|x|\leq\pi$ and $|y|\leq\pi$.

$\nabla\phi(x,y) = \langle \cos(x)\sin(y), \sin(x)\cos(y)\rangle$

403. Find the gradient field $\mathbf F = \nabla\phi$ for the potential function $\phi(x,y,z) = e^{-z}\sin(x+y)$

$\mathbf F = e^{-z}\left\langle \cos(x+y), \cos(x+y), -\sin(x+y)\right\rangle$

## Line Integrals

420. Evaluate $\int_C(x^2+y^2)ds$ if C is the line segment from (0,0) to (5,5)

$\frac{250\sqrt2}3$

421. Evaluate $\int_C(x^2+y^2)ds$ if C is the circle of radius 4 centered at the origin

$128\pi$

422. Evaluate $\int_C(y-z)ds$ if C is the helix $\mathbf r(t) = \langle 3\cos(t),3\sin(t),t\rangle,\ t\in[0,2\pi]$

$-2\sqrt{10}\pi^2$

423. Evaluate $\int_C \mathbf F\cdot d\mathbf r$ if $\mathbf F =\langle x,y\rangle$ and C is the arc of the parabola $\mathbf r(t) = \langle 4t,t^2\rangle,\ t\in[0,1]$

$17/2$

424. Find the work required to move an object from (1,1,1) to (8,4,2) along a straight line in the force field $\displaystyle\mathbf F = \frac{\langle x,y,z\rangle}{x^2+y^2+z^2}$

$\ln(2\sqrt7)$

## Conservative Vector Fields

Determine if the following vector fields are conservative on $\mathbb R^2.$

440. $\langle -y, x+y \rangle$

No

441. $\langle 2x^3+xy^2, 2y^3+x^2y\rangle$

Yes

Determine if the following vector fields are conservative on their respective domains in $\mathbb R^3.$ When possible, find the potential function.

442. $\langle y,x,1\rangle$

$\phi(x,y,z)=xy+z$

443. $\langle x^3,2y,-z^3 \rangle$

$\phi(x,y,z)=(x^4+4y^2-z^4)/4$

## Green's Theorem

460. Evaluate the circulation of the field $\mathbf F=\langle 2xy,x^2-y^2\rangle$ over the boundary of the region above y=0 and below y=x(2-x) in two different ways, and compare the answers.

$0$

461. Evaluate the circulation of the field $\mathbf F=\langle 0,x^2+y^2\rangle$ over the unit circle centered at the origin in two different ways, and compare the answers.

$0$

462. Evaluate the flux of the field $\mathbf F=\langle y,-x\rangle$ over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.

$0$

## Divergence And Curl

480. Find the divergence of $\langle 2x, 4y, -3z\rangle$

$3$

481. Find the divergence of $\displaystyle\frac{\langle x,y,z\rangle}{1+x^2+y^2}$

$\displaystyle\frac{x^2+y^2+3}{(1+x^2+y^2)^2}$

482. Find the curl of $\langle x^2-y^2, xy, z\rangle$

$\langle 0,0,3y\rangle$

483. Find the curl of $\langle z^2\sin(y), xz^2\cos(y), 2xz\sin(y)\rangle$

$\mathbf 0$

484. Prove that the general rotation field $\mathbf F = \mathbf a\times\mathbf r$, where $\mathbf a$ is a non-zero constant vector and $\mathbf r = \langle x,y,z\rangle$, has zero divergence, and the curl of $\mathbf F$ is $2\mathbf a$.

If $\mathbf a = \langle a_1,a_2,a_3\rangle$, then

$\mathbf F = \mathbf a \times \mathbf r = \langle a_2z-a_3y, a_3x-a_1z, a_1y-a_2x\rangle = \langle f,g,h\rangle$, and then

$\nabla\cdot\mathbf F = \mathbf f_x + \mathbf g_y + \mathbf h_z = 0+0+0=0,$

$\nabla\times \mathbf F = \langle h_y - g_z, f_z-h_x, g_x-f_y\rangle = \langle a_1 + a_1, a_2+a_2, a_3+a_3\rangle = 2\mathbf a.$

## Surface Integrals

500. Give a parametric description of the plane $2x-4y+3z=16.$

$\langle u,v,(16-2u+4v)/3\rangle,\ u,v\in\mathbb R$

501. Give a parametric description of the hyperboloid $z^2=1+x^2+y^2.$

$\langle\sqrt{v^2-1}\cos(u),\sqrt{v^2-1}\sin(u),v\rangle,\ u\in[0,2\pi],\ |v|\geq1$

502. Integrate $f(x,y,z) = xy$ over the portion of the plane z=2−xy in the first octant.

$2/\sqrt3$

503. Integrate $f(x,y,z) = x^2 + y^2$ over the paraboloid $z=x^2+y^2,\ z\in[0,4].$

$\frac{(391\sqrt{17} + 1)\pi}{60}$

504. Find the flux of the field $\mathbf F = \langle x,y,z\rangle$ across the surface of the cone
$z^2=x^2+y^2, \ z\in[0,1],$
with normal vectors pointing in the positive z direction.

$0$

505. Find the flux of the field $\mathbf F = \langle -y,z,1\rangle$ across the surface
$y=x^2, \ z\in[0,4],\ x\in[0,1],$
with normal vectors pointing in the positive y direction.

$-10$

## Stokes' Theorem

520. Use a surface integral to evaluate the circulation of the field $\mathbf F = \langle x^2 - z^2, y, 2xz \rangle$ on the boundary of the plane $z = 4-x-y$ in the first octant.

$\frac{-128}{3}$

521. Use a surface integral to evaluate the circulation of the field $\mathbf F = \langle y^2,-z^2,x \rangle$ on the circle $\mathbf r(t) = \langle 3\cos(t),4\cos(t),5\sin(t) \rangle.$

$15\pi$

522. Use a line integral to find $\iint_S(\nabla\times F)\cdot\mathbf n dS$
where $\mathbf F = \langle x,y,z \rangle$, $S$ is the upper half of the ellipsoid $\frac{x^2}4 + \frac{y^2}9 + z^2 = 1$, and $\mathbf n$ points in the direction of the z-axis.

$0$

523. Use a line integral to find $\iint_S(\nabla\times F)\cdot\mathbf n dS$
where $\mathbf F = \langle 2y,-z,x-y-z \rangle$, $S$ is the part of the sphere $x^2 + y^2 + z^2 = 25$ for $3 \leq z \leq 5$, and $\mathbf n$ points in the direction of the z-axis.

$-32\pi$

## Divergence Theorem

Compute the net outward flux of the given field across the given surface.

540. $\mathbf F = \langle x, -2y, 3z \rangle$, $S$ is a sphere of radius $\sqrt 6$ centered at the origin.

$16\sqrt6\pi$

541. $\mathbf F = \langle x, 2y, z \rangle$, $S$ is the boundary of the tetrahedron in the first octant bounded by $x+y+z=1$

$2/3$

542. $\mathbf F = \langle y+z, x+z, x+y \rangle$, $S$ is the boundary of the cube $\{(x,y,z)\mid |x|\leq 1, |y|\leq 1, |z|\leq 1\}$

$0$

543. $\mathbf F = \langle x, y, z \rangle$, $S$ is the surface of the region bounded by the paraboloid $z=4-x^2-y^2$ and the xy-plane.

$24\pi$

544. $\mathbf F = \langle z-x, x-y, 2y-z \rangle$, $S$ is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.

$-224\pi$

545. $\mathbf F = \langle x, 2y, 3z \rangle$, $S$ is the boundary of the region between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$ and cut off by planes $z=0$ and $z=8$

$144\pi$