8) The vector field Vp[x, y] = F =- has a potential function p[x, y] = a) (x² + y²)³2 b) d) Vx? +y? e) F[X] has no potential function.

Q: The gradient vector field of for,y)= In +2y) is 2. T True A False

A: The gradient vector field of a function, is defined by the formula below. , where are the partial…

Q: (x, y) (1.0) (2, 2) (3,0) (0, 1) (-2,2) (0, 3) F(x, y) (0, 1) (-2,2) (0.3) (-1,0) (-2,-2) (-3,0)…

A: This is a question of vector calculus.

Q: f is a potential function for the vector field F = (2x - 2yz, 3y2 - xz, -2xy) so what is the value…

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Q: If f is a potential function for the vector field F = (1+2ry+2:², -1+², 4x) then the value of f(-1,…

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Q: Find the flux of the vector field F(x, y, z) = ri across the outward oriented sphere %3D

A: In this question, we find the flux of the vector field f(x,y,z)=xi The surface is a sphere of radius…

Q: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

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Q: If f is a potential function for the vector field F = (y + 2xz, x - 3y²z², x² - 2y³z), then the…

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Q: 11) If f is a potential function for the vector field F = (-2y + 2xyz, -2x + x²z, x² y + 8z), then…

A: To find- If f is a potential function for the vector field F = -2y + 2xyz, -2x + x2z, x2y + 8z, then…

Q: a) Is the vector field F = (a2 - yr, y?- xy) path independent (conservative)? %3D O True O False b)…

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Q: 2) Find the divergence of the vector field F(r, y, z) = (re") i + (ye") j+ (ryz) k %3D

A: We have to find divergence

Q: Is the vector field F = y? – 2x², 2xy + 3| conserva Yes No yz Is the vector field F y? – x2 con Vy?…

A: According to the guidelines issued by the company we are supposed to answer only first question.…

Q: Question 4 a) Given that o = x + y +3xyz, find gradø and hence evaluate Igradol at the point…

A: a) From the given problem: φ=x3+y3+3xyz So,…

Q: 2. Given vector field F= (-2y, z-2x, y + z), find the work done by ♬ on a particle that moves along…

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Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: 24. For a constant vector E, find the constant k such that V(FE) = KË: C) km-l D) - A)k-0 B)k-1 E)…

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Q: 1. Find a potential function for the vector fields a) b)

A: As per the company's guidelines, we are supposed to solve the first question for you, kindly repost…

Q: 1- a) c) −2ñi + (2ñ + 1)j b) 2πi + (2π − 1)j - 4пi – (2n + 1)j 2лi + (2π + 1)j 4ñi + (2ñ + 1)j Leave…

A: Given:The given function is .To Find:We have to determine the gradient vector of  at the point…

Q: Show that curl(F) = 0 and that, SF· dr = 27 where C is the circle of radius 1 centered at the origin…

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Q: a) Determine whether or not the vector field F 8yz, – 6 cos(y) – 8xz, – 8xy + 10z > - is…

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Q: h) Given vector field F (x, y, z) = (2x + 2y + 4z) i + (2y + 2x – 32) j + (2z – 3y + 4x) k. (a)…

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Q: a)Suppose, w = d(z,9) + ip(z, v) defines a complex potential function. Terify whether w is analytic…

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Q: 4. Show the vector field Vu x Vv, where u and v are scalar fields, is solenoidal.

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Q: If f is a power function for the vector field F = (2xz, −3z, x2 −3y+12z2), then the value of f(−3,…

A: We have F=2xz, -3z, x2-3y+12z2 Now, F=∇f=∂f∂x, ∂f∂y, ∂f∂z=2xz, -3z, x2-3y+12z2 So, ∂f∂x=2xz…

Q: 1) Given the vector fields F1 = x²2 and F2 = xâ+ yŷ + z2, calculate the divergence and curl of F1…

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Q: Given the vector field F(x,y,z) = (y²,2xy-e²¹) and the path C given by r(t) = cos(t)i + sin(t)];…

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Q: 4. If ricost cosh t+j sint sinh t, where i and j are constant orthogonal vectors prove that (a)…

A: Given r = i cost cosht + j sint sinht

Q: 9. Find the flux of the vector field B(x, y,=) shown in the left figure through the square S of - yỉ…

A: The general form of a vector valued function in three dimensions is given by…

Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

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Q: Let A = (A, A,y, A2) and B = (B,, By, B2) be two arbitrary vector fields, prove the following…

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