Compute the line integral of the vector field F = (3zy1, 4z, -y) over the path c(t) = (e*, e', t) for –3 < t < 3 SeF - ds =

Q: What is the line integral of a vector field?

A: To define the line integral of a vector field.

Q: Show that work performed by a constant force field F over any path C ~ from P to Q is equal to F •…

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Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: Compute the work done by the vector field F(x, y, z) = (,2, 2yz) when moving a particle: 1+? > a)…

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Q: Find the total work done in moving particle in a free field given by F = 3xyi – 5zj + 10xk along the…

A: The total work done by a vector field, F→ moving a particle along the curve r→(t) is calculated…

Q: Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, –6z – 32²y²)…

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Q: Compute the curl of the following vector field. F = = (3z2 sin y, 3xz2 cos y, 6xz sin y)

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Q: Compute the scalar line integral x – yz ds where C is the line seg- ment from (0, 1, 1) to (1,0, 1).

A: Given  ∫C x-yz   ds    where C is line segment from (0,1,1) to ( 1,0,1)

Q: Suppose you need to integrate the vector fieldv – (3 cos z, ef sin y)) along the curve parameterised…

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Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

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Q: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

Q: Find the line integral F.dr of the vector field F=ax k over a curve C defined by a C rectangle in…

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Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

Q: Find the work done by the vector field F = in moving a particle around the circle x2 +y2 = 4 from…

A: The given vector field is F=x-y, xy. The particle is moving around the circle x2+y2=4 from 2,0 to…

Q: Compute the line integral of the vector field F=⟨3zy-1 ,4x,−y⟩ over the path c(t)=(et,et,t)for…

A: To Compute the line integral of the vector field F=⟨3zy-1 ,4x,−y⟩ over the path c(t)=(et,et,t)for…

Q: 3 , find a potential function for the vector field F by inspection or show that one does not exist.…

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Q: Choose a correct potential function for the conservative vector fieldF = (3x-y+1, –x-4y-2).

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Q: Use the Fundamental Theorem for Line Integrals to find the work done by the conservative vector…

A: If F(x,y,z) is a conservative field then there exist a scalar function f, such that ∇f = F. By…

Q: Find the gradient vector field of f бу f(x, у, 2) = X COS Z (бху) (6y бу -sin бх -sin бу i cos k…

A: Given,

Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: 2. Show that V(V × F) = 0, here vector field is twice differentiable. %3D

A: This question is from vectors.

Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

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Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

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Q: Show that the line integral is independent of path. | 2xe Ydx + (2y - x²e)dy, Cis any path from (1,…

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Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

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Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

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Q: Find the gradient field associated with the function φ(x, y, z) = xyz.

A: To find the gradient field associated with the function  φ(x, y, z) = xyz.

Q: Find a potential functioon for the conservative vector field given by y3/2 3 xy + z, 2 sin z 2.x…

A: According to question,Given vector field,F→x, y, z = 2xx2 + cos z-y322x, -32xy+z, -sin zx2 + cos z+y

Q: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: Calculate the work done in moving a particle in the force field F = 3x2î+ (2xz-y)ĵ+ z k along a…

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Q: Suppose a particle travels clockwise on a triangular path connecting the points (0,0), (3,1), and…

A: greens theorem

Q: Evaluate line integral of vector field F(x, y, z) = y sin(x²) i+ xyz j+ (x +y + z) k along line…

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Q: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

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Q: (2) Find the line integral of the vector field F(x, y) = (-1,3) along the path from (2, 3) to (2,…

A: The vector field F→=-1, 3To find line integral of vector field along the path 2, 3 to 2, 5

Q: Find the curl of the vector field F=(2xyz, -xy,-z^2)

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Q: Suppose you need to integrate the vector field v = (3 cos z, e+, sin(y)) along the curve…

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Q: Determine the gradient of the give scalar field V = e(2*+3y) cos 5z at R(-0.1, 0.2, 0.4).

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Q: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

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Q: Find the flux out of the curve bounded by the top half of = arccos(√√²+² ) and y=0 with vector field…

A: As per the question we are given a bounded curve C and a 2d vector field F And we have to find the…

Q: Consider the conservative vector field 4 8 F (x, y, z) 3y cos (372), 37y sin (372) – a² > 2xz, 2x +…

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Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: 6. Show that the system with vector field -x + sin(y) F(x, y) : -y+ cos(x), admits no periodic…

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Q: Show that the line integral is independent of path. 2xe-Ydx + (2y - x2e-Y)dy, Cis any path from (1,…

A: The product rule of differentiation states that d dxfg=fdgdx+gdfdx, where f, g are functions of x.…

Q: The integral of the vector field F(x, y) = (cos |x|, sin y|) A = (-2,0) and final point B (0, 2),…

A: Correct answer is (B) sin(2) - cos(2) + 1 Explanation:

Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

Q: Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical…

A: