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Let F(x, y, z) = (2xsin(ay) - e3z)i + 2x+cos(2y)j + bre3zk. a) Find the values...
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin...
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
7. (6pts) Consider F(x, y, z) = (y2 + z cos x)i + (3xy2 + 1)j...
7. (6pts) Consider F(x, y, z) = (y2 + z cos x)i + (3xy2 + 1)j + sin æk. Show that F is a conservative vector field and then compute SF. dr where C is any curve from (0,0,1) to (0,2,3).
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please find (2y+ye 5. С:/(r)-(cost.sin 1,1). Osis". dy b. Please to prove that F is a conservative vector field: ye". c....
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please find (2y+ye 5. С:/(r)-(cost.sin 1,1). Osis". dy b. Please to prove that F is a conservative vector field: ye". c. Please find J2xye d. Please find the potential function fx, y, z) such that F Vf e. Use the part (d) to evaluate F dr along the given curve C. f. Please find curlF g. Please find curlF
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient...
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient field and then find the function f(x,y) such that F = VS. Use it to evaluate line integral SF. dr where the curve C is the arc of the circle 12 + y2 = 4 from (2,0) to (0,2)
Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey...
Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey + e*, ye*). (a) (5 points) Compute V F(x, y, z). (b) (10 points) Find a potential function for F or explain why none exists. (c) (5 points) Find ScF. dr, where C is the curve consisting of the line segments from (0,0,0) to (1,2,0), from (1,2,0) to (1,2, 1), and from (1, 2, 1) to (1,2,2).
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S...
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S = F (potential function). (5) Use the results in part(a) to cakulae ( F. ds along C which the curve y = a* from (0,0) to (2,16). (2) Use Green's Theorem to evaluate 1. F. ds. F(1,y) =(yº+sin(26))i + (2xy2 + cos y)and C is the unit circle oriented counter clockwise (6) Evaluate the surface integral || 9. ds. F(x,y,z) = xi +++where S...
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z...
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z +...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
(1 point) Let F (72+72) i + (2y +62 + 6 sin(y*)) 3+ (2x + 6y...
(1 point) Let F (72+72) i + (2y +62 + 6 sin(y*)) 3+ (2x + 6y + 2e=") R. (a) Find curl F. curl F = <0,0,0> (b) What does your answer to part (a) tell you about Sc F. dr where is the circle (2 – 30)2 + (y - 35)2 = 1 in the ty-plane, oriented clockwise? SF. dr = 0 (c) If C is any closed curve, what can you say about SF. dr? SoF dr =...
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
7. (6pts) Consider F(x, y, z) = (y2 + z cos x)i + (3xy2 + 1)j + sin æk. Show that F is a conservative vector field and then compute SF. dr where C is any curve from (0,0,1) to (0,2,3).
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please find (2y+ye 5. С:/(r)-(cost.sin 1,1). Osis". dy b. Please to prove that F is a conservative vector field: ye". c. Please find J2xye d. Please find the potential function fx, y, z) such that F Vf e. Use the part (d) to evaluate F dr along the given curve C. f. Please find curlF g. Please find curlF
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient field and then find the function f(x,y) such that F = VS. Use it to evaluate line integral SF. dr where the curve C is the arc of the circle 12 + y2 = 4 from (2,0) to (0,2)
Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey + e*, ye*). (a) (5 points) Compute V F(x, y, z). (b) (10 points) Find a potential function for F or explain why none exists. (c) (5 points) Find ScF. dr, where C is the curve consisting of the line segments from (0,0,0) to (1,2,0), from (1,2,0) to (1,2, 1), and from (1, 2, 1) to (1,2,2).
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S = F (potential function). (5) Use the results in part(a) to cakulae ( F. ds along C which the curve y = a* from (0,0) to (2,16). (2) Use Green's Theorem to evaluate 1. F. ds. F(1,y) =(yº+sin(26))i + (2xy2 + cos y)and C is the unit circle oriented counter clockwise (6) Evaluate the surface integral || 9. ds. F(x,y,z) = xi +++where S...
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
(1 point) Let F (72+72) i + (2y +62 + 6 sin(y*)) 3+ (2x + 6y + 2e=") R. (a) Find curl F. curl F = <0,0,0> (b) What does your answer to part (a) tell you about Sc F. dr where is the circle (2 – 30)2 + (y - 35)2 = 1 in the ty-plane, oriented clockwise? SF. dr = 0 (c) If C is any closed curve, what can you say about SF. dr? SoF dr =...
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