Show that the vector field F=2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds along the boundary of the square with vertices (1,0), (2,0), (2,1) and (1,1) taken counter clockwise.

Q: Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y).   Enter the exact…

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Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: Find the conservative vector field for the potential function by finding its gradient.

A: Gradient for function h is given by , where are the partial derivatives. Partial derivative…

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: Calculate the divergence of the vector field. F(x, y) = 3x°i – 2x sin(xy)j div(F) =

A: Given vector function F(x,y)=3X3 i -2x sin(xy) j divF=i∂∂x+j∂∂y+k∂∂z.F

Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

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Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

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

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Q: Find the conservative vector field for the potential function by finding its gradient. 9 (r,y, 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: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

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Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

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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: Compute the line integral of the vector field F = (3zy1, 4z, -y) over the path c(t) = (e*, e', t)…

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Q: Determine if the vector field F is conservative. F= cos x cos yi + sin x sin yj - sec |- sec zk O…

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Q: etch the vector field F(x, y) = xi + yj and calculate the line integral of F along the line segment…

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Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

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Q: Find the work W done by the vector field ⟨2x+yx,x2+4⟩ on a particle moving along the boundary of the…

A: To find the workdone by the vector field.

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: Show that F of F. = (2xy + z³)i+x²+3xz²k is conservative force field. Find the scalar potential

A: *By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: Calculate the integral line of vector field F(x,y) around the circle x²+y²=1 oriented…

A: Given that Fx,y=-2yx2+y24+2x,2xx2+y2/4+xy+1. The the integral line of vector field F(x,y) around the…

Q: Find the divergence of the vector field F div F -6z sin (a + 7) %3D

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Q: Consider the conservative vector field 8. 4 F(x, y, 2) = (- 2xa, - 3y*cos (32)3ny³sin (3nz) – x²). -…

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Q: Compute the flux of the vector field F(x,y, z) = yi– 2xyk over the triangle T with vertices (1,0,0),…

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Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

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Q: Calculate the line integral of the vector field F = -2i+ 4jalong the line from the point (3,0) to…

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Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

Q: 3. Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, -6z –…

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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: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

A: Given F=xi+yj+zk ,-3<=x<=3,r=1

Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

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Q: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

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Q: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

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Q: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

Q: he line integral of the vector field F= over the curve C which is the quarter of the rcle from…

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Q: 6. Show that F = [y? cos x + z3, 2y sin x – 4, 3xz2 + 2] is a conservative force field. Find the…

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Q: Find the conservative vector field for the potential function by finding its gradient. h(x, y, z) =…

A: We have to find the conservative vector field.

Q: Compute the outward flux of the vector field F = (5xy, 5y²) for the region enclosed by the curves y…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Calculate F=, around the triangle with vertices: (0,0,2), (2,0,0) and (0,2,0). Can you use Stoke's…

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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: Let F(x, y, z) =⟨3x2y+az, x3,3x+ 3z2⟩ be a vector field. For what values of a is F conservative?

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Q: Consider the conservative vector field given by: F (x, y) = (x - ycos (x), y - sin (x)) A potential…

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Q: Consider a conservative vector field: F(x, y) = (y cosx + y2, sinx + 2xy – 2y) Find a potential…

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Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution