# ( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy

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Download "( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy"

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1 . Let F = ( yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F F = yz x B. curl F + ( 3xz) y + ( 9xy) = = 0 z F = i j k x y z yz 3xz 9xy = 6xi 8yj+ 2zk C. div curl F F = 6x x + ( 8y) y + ( 2z) = = 0 z

2 2. Let F( x, y,z) = ( 5xz 2, 6xyz, 6xy 3 z) be a vector field and f ( x, y,z) = x 3 y 2 z. Find f ( f x, f y, f z ) = 3x 2 y 2 z, 2x 3 yz, x 3 y 2 Find F i j k y x y y y z 5xz 2 6xyz 6xy 3 z 3y 2 = 6x z 6xy i + 6y3 z 0xz j + [ yz 0]k Find F f = 6x i j k 5xz 2 6xyz 6xy 3 z ( 4 y 3 z + 2x 4 y 4 z 2 )i + 5x 4 y 2 z 2 8x 3 y 5 z 2 3x 2 y 2 z 2x 3 yz x 3 y 2 j + 0x 4 yz 3 8x 3 y 3 z 2 k Find F f ( 5xz 2, 6xyz, 6xy 3 z) 3x 2 y 2 z, 2x 3 yz, x 3 y 2 5x 3 y 2 z 3 + 2x 4 y 2 z 2 6x 4 y 5 z

3 3. Let F= ( 0xyz + 3sin x, 5x 2 z, 5x 2 y). Find a function f so that F= f, and f ( 0,0,0) = 0. F = ( 0xyz + sin x, 5x 2 z, 5x 2 y) f x = 0xyz + 3sin x f = 5x 2 yz 3cos x f y = 5x 2 z f = 5x 2 yz + C 2 x,z f z = 5x 2 y f = 5x 2 yz + C 3 x, y f = 5x 2 yz 3cos x + C = 3 + C = 0 C = 3 f 0,0,0 f = 5x 2 yz 3cos x + 3

4 4. For each of the following vector fields F, decide whether it is conservative or not by computing the curl F. Type in a potential function f ( that is, f = F). If not conservative, type N. A. F( x, y) = ( 6x + 4y)i + ( 4x + 2y) j M = -6x + 4y and N = 4x + 2y Take the partial derivative in terms of x and y. ( 6x + 4y) y = 4 = 4x + 2y x The field is conservative. dx = 8x 2 + 4yx + k( y) M ( 8x 2 + 4yx + k( y) ) y = 4x + k ' y f ( x, y) = 8x 2 + 4yx + y 2 B. F( x, y) = 8yi 7xj ( 8y) y = 8 ( 7x) x = 7 Not conservative. = 8xi 7yj + k ( 8x) = 0 x y C. F x, y,z 7y It is conservative. + ( 0,0,) F= 8x, 7y, 0 f ( x, y,z) = 4x y2 + z = 4x + 2y k '( y) = 2y D. F = ( 8sin y)i + ( 8y 8x cos y) j N x M y f = 8xsin y + 4y 2 = 8cos y ( 8cos y ) = 0 E. F( x, y,z) = 8x 2 i + 4y 2 j + z 2 k f = 8 3 x y3 + z3 3

5 5. Let C be the curve which is the union of two line segments, the first going from ( 0,0) to ( 3,4 ) and the second going from ( 3,4 ) to ( 6,0). Compute the integral 3dy 4 dx. ( 6,0) ( 4,3) ( dx,dy) = 4x + 3y ( 0,0) = 24 C C

6 6. Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine for the following vector field F whether the line integrals F dr are positive, negative, or zero and type P, N, or Z as appropriate. C x = r cosθ dr = dx,dy y = r sinθ dθ = y, x A. F = the radial vector field = xi + yj : ( x, y) ( y, x)dθ = 0 B. F = the circulating vector field= yi + xj ( y, x) ( y, x)dθ = r 2 dθ therefore it is positive C. F = the circulating vector field = yi xj ( y, x) ( y, x)dθ = r 2 dθ which is negative D. F = the constant vector field=i+j The field is constant and a constant field is always conservative.

7 7. Suppose that f ( x, y,z) = 2xyze x2 i + ze x2 j + ye x2 k. If f ( 0,0,0) = 3, find f ( 2,2,6). f x = 2xye x2 f = yze x2 + C y,z f y = ze x2 f = yze x2 + C 2 x,z f z = ye x2 f = yze x2 + C 3 ( x, y) = yze x2 + C = C = 3 f x, y,z f 0,0,0 f ( 2,2,6) = 2e 4 3

8 8. Evaluate the line integral x 4 z ds, where C is the line segment from 0, 2, 5 L :( 0,2,5) + t ( 7,6, 4) = ( x, y,z) ds = t =0 ( dx) 2 + ( dy) 2 + ( dz) 2 = ( 7,6, 4) dt = 0dt ( 7t) 4 5 4t 0dt t t 6 0 C 4 = to 7,8,.

9 9. Evaluate the line integral 5xy 2 ds, where C is the right half of the circle x 2 + y 2 = 36. C x = 6cosθ y = 6sinθ, θ from π 2 to π 2 π 2 θ = π ( 6 3 )sin 2 θ cosθ ( 6dθ ) ds = 6dθ (arc length) π 2 sin 2 θd sinθ θ = π 2 3 sin3 θ π 2 π 2 = ( 5) ( 6 3 )( 4)

10 0. Suppose C is any curve from 0,0,0 F( x, y,z) = ( 4z + 5y)i + ( 3z + 5x) j + 3y + 4x F is conservative. f = F f = 4zx + 5xy + 3yz (,,) 4zx + 5xy + 3yz 0,0,0 = 2 to (,, ) and k. Compute the line integral F dr. C

11 . Let C be the positively oriented square with vertices ( 0,0), (,0 ), (,), ( 0,). Use Green's Theorem to evaluate the line integral 4y 2 x dx + 8x 2 ydy. ( 4y 2 x, 8x 2 y) ( dx,dy) x=0 y=0 ( 6xy 8xy)dx dy 8 x dx ydy = 8 x=0 y=0 4 = 2 C

12 2. Let C be the positively oriented circle x 2 + y 2 =. Use Green's Theorem to evaluate the line integral 8y dx + x dy. ( 8y, x) dr r =0 7π θ =0 C ( 8)r dr dθ

13 3. Let F = 2xi + yj and let n be the outward uni normal vector to the positively oriented circle x 2 + y 2 =. Compute the flux integral F nds. Method You can use Gauss' Divergence Theorem F n ds = F da F nds = ( 2 + )da = 3π S C. C S Method 2 ( 2x, y) ( x, y)ds = 2x dy + y dx x = cosθ y = sinθ ds = rdθ ds = dθ because the radius is. dx = ydθ dy = cosθdθ ( y,2x) dr S N x M y = 3 π = 3 the area ( )2 = 3π

14 4. Use Green's Theorem to compute the area inside the ellipse x2 3 + y2 2 9 =. 2 Use the fact that the area can be written as dx dy = D D Q x P y = We need P, Q Q x P y = We choose P = 0 and Q = x. ( P,Q) dr = x dy x y2 9 2 = x = 3cosθ θ =0 y = 9sinθ dy = 9cosθdθ θ =0 θ =0 3( 9)cos 2 θ dθ 3 9 cos2θ + 2 dθ 3( 9) 2 θ = ( 3) ( 9)π = 57π 0 P dx + Q dy. D

15 5. Let F = 3yi + 2xj. Use the tangential vector form of Green's Theorem to compute the circulation integral F dr where C is the positively oriented circle x 2 + y 2 = F dr = F Tds = 2 ( 3)dA C x 2 + y 2 = 5π

16 6. Evaluate F ds where F = 3xy 2, 3x 2 y, z 3 and M is the surface of the sphere M of radius 6 centered at the origin. F ds = F nds = FdV M B 6 θ =0 3y 2 + 3x 2 + 3z 2 dv dθ 2 π φ = sinφ dφ 6 ρ=0 = B ρ 4 dρ

17 7. Find the outward flux of the vector field F= ( x 3, y 3, z 2 ) across the surface of the region that is enclosed by the circular cylinder x 2 + y 2 = 9 and the planes z = 0 and z = 4. F ds = 3x 2 + 3y 2 + 2z dv y θ =0 θ =0 3 r =0 3 r =0 3r 3 r x 2 + y 2 =9 z=0 4 z=0 4 dz dr dθ 2z dz dr dθ + ( ) ( 4 2 ) = 630π

18 8. Let F= ( y 2 + z 3, x 3 + z 2, xz). Evaluate F ds for each of the following regions W : F = x = x A. x 2 + y 2 z 6 θ =0 6 r =0 6 z=r 2 sinθ 0 = 0 r cosθ dzr dr dθ w B. x 2 + y 2 z 6, x 0 π 2 θ = π 2 6 r =0 6 z=r 2 r 2 cosθ dz dr dθ π 6 2 sinθ π ( 6 r 2 )r 2 dr r = r 3 5 r = C. x 2 + y 2 z 6, x 0 A = B + C C = B

19 9. Use the divergence theorem to find the outward flux of the vector field = 4x 2 i + 3y 2 j + 5z 2 k across the boundary of the rectangular prism: F x, y,z 0 x, 0 y 5, 0 z 2. F = 8x + 6y + 0z x=0 x=0 x= y=0 5 y=0 2 z=0 8x + 6y + 0z dz dydx x + 2ydydx xdx

20 20. Let S be the part of the plane 4x + y + z = 3 which lies in the first octant, oriented upward. Find teh flux of the vector field F = i + 4 j + 2k across the surface S. The surface given by z = 4x y + 3 is bounded by n = ( 4,,) n = 4,, ds = ds = 3 4 x=0 3 4 x=0 3 8 ( z x ) 2 + ( z y ) 2 + da ( 4) x y=0 3 4 x y=0 2 + da = 8dA ( 4,,) (,4,2 ) 8 da 0dydx x dx x= x 2x = ,0,0, ( 0,3,0 ), and ( 0,0,3).

21 2. Find directly the flux of F( x, y,z) = ( 3xy 2, 3x 2 y, z 3 ) out of the sphere of radius 3 centered at the origin, without the aid of Gauss' Divergence Theorem. The flux is given by the integral: 3 5 b d f ( θ,φ)dθ dφ. F = 3y 2 + 3x 2 + 3z 2 = 3ρ 2 θ =0 3ρ π φ =0 3 π a=0 3 ρ=0 π 3ρ 2 ρ 2 sinφ dρ dφ dθ c=0 sinφ dθ dφ π 5 sinφ dθ dφ a=0 c=0 a = 0, b = π, c = 0, d = Find f ( θ,φ) θ =0 F nds π 2 φ =0 3 2 sinφdφdθ ( 3xy 2, 3x 2 y, z 3 ) ( 3,3,3 ) 3 f ( θ,φ) = 6cos 2 θ sin 2 θ sin 5 φ + cos 4 φ sinφ a c θ =0 π 2 φ =0 729π ( 3sin 4 φ cos 2 φ sin 2 θ + 3sin 4 φ cos 2 φ sin 2 θ) + cos 4 φ sinφ dφ dθ

22 22. Let F = ( 2x, 2y, 2x + 2z). Use Stokes' Theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (,0,), ( 0,,0 ) and ( 0,0,). In particular, compute the unit normal vector of the plane spanned by the points (,0,), ( 0,,0 ), and ( 0,0,) and the curl of F as well as the value of the integral: Find the normal vector of the plane. n = n = 0,, 2 i j k = ( 0,,) Find F i j k x y z 2x 2y 2x + 2z = ( 0, 2,0) FdS = ( 0, 2,0) ( 0,,)dA where ds = 2dA 2 2 x=0 x=0 x y=0 dy dx ( x)dx 2 x 2 x2 0 =

23 23. Use Stoke's Theorem to evaluate F dr C where F( x, y, z) = xi + yj + 3( x 2 + y 2 )k and C is the boundary of the part of the paraboloid where z = 64 x 2 y 2 which lies above the xy-plane and C is oriented counterclockwise when viewed from above. F nds F nda D x x 2 + y 2 64 y z dx dy ( 2x) ( 6y) + 2y( 6x) + 0dx dy 0dA = 0 x 2 + y 2 64

24 where zk and S is the part of the paraboloid 24. Use Stoke's Theorem to evalute curlf ds S F( x, y,z) = yzi + xzj+ 8 x 2 + y 2 z = x 2 + y 2 that lies the cylinder x 2 + y 2 =, oriented upward. F dr = yz, xz, 8r 2 z S z= x 2 + y 2 = dx,dy,0 ( yz, xz, 8r 2 z) ydθ, xdθ, 0 z= x 2 + y 2 = y 2 + x 2 dθ = cos 2 θ + sin 2 θdθ = dθ = θ =0 0 0

25 25. Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x 2 + y 2 = 49, 0 z, and a hemispherical cap defined by x 2 + y 2 + ( z ) 2 = 49, z. For the vector field F= ( zx + z 2 y + 5y, z 3 yx + 5x, z 4 x 2 ), compute ( F) ds in any way you like. M FdS = F dr M M ( zx + z 2 y + 5y, z 3 yx) ( y, x)dθ where z = 0 x 2 + y 2 = 49 0

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