Calculus/Multivariable and differential calculus:Exercises

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Multivariable and differential calculus:Exercises

Parametric Equations[edit]

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[edit]

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)
2-2sin-t.svg
23. r^2 = 4\cos(\theta)
R-square-eq-4cos-t.svg
24. r=2\sin(5\theta)
2sin-5t.svg

Sketch the following sets of points.

25. \{(r,\theta):\theta=2\pi/3\}
Polar-set-answer-1.svg
26. \{(r,\theta):|\theta|\leq\pi/3\mbox{ and }|r|<3\}
Polar-set-answer-2.svg

Calculus in Polar Coordinates[edit]

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
2-plus-cos-t.svg
43. The region inside the petals of the rose 4\cos(2\theta) and outside the circle r=2
8\pi/3 + 4\sqrt3
4cos-2t-and-2.svg

Vectors and Dot Product[edit]

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)
Plane-intercepts-2-3-4.svg
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

Failed to parse (unknown function '\begin'): \begin{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[edit]

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)

Failed to parse (unknown function '\begin'): \begin{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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Failed to parse (syntax error): \displaystyle f_s = \frac{-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[edit]

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[edit]

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)

Saddle at (0,0)

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Radial vector field.svg
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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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