Consider the vector field v(x) i-j+ 6k. a) Calculate fvo dr, where I is the straight line from (4,0,4) to (0,4,8). b) Calculate fu• dr where C'is the curve from (4,0,4) to (0,4, 8) in the first quadrant given by x? + 4y? = 22 , x +z = 8.

Q: Sketch the following gradient vector field, F(x, y, z) = 2xi - 2yj – 2xk. Then find the vector…

A: The given problem is to sketch the given vector field and also to find the gradient of the given…

Q: Compute the flux of the vector field F = 5xi + 2yj through the cylindrical surface shown below, if a…

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Q: Find the flux of the vector field V(x, y, z) = 3xy²i + 5x²yj + 2z³k out of the unit sphere. Flux =

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Q: 5) By using Stokes' Theorem, find the work performed by the vector field Ex² + 4xy³i+ y²xk,on a…

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Q: xi + yj x2 + y2 F(x, y) = k-

A: Topic- vector integration

Q: 2. Verify that the vector field F(1, y, 2) = sec? y+ -32 2.r tan y + e csc r, 3e -32 cot x + sec z…

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Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: Compute the flux of the vector field F 3.ri + 3yj through the cylindrice surface shown below, if a=3…

A: To calculate the flux of the vector field.

Q: F = (x² +y², y² +z?,.

A: on solving this we get the

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

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Q: Compute the curl of the vector field F = - (x + 3y) i + (y – 3z) } – (z + 3x) k curl

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Q: Compute the flux, || Foñds, across the slanted face of the tetrahedron 2 = 4-1- y bounded in the…

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Q: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

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Q: For the vector field F(x, y, z) = (15x2y + 2xz)i + (5x3 + 35y4)j + x2k, show that F is conservative…

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Q: ) Compute the flux of the vector field F = 5xi + 2yj through the cylindrical surface shown below, if…

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Q: Determine whether the vector field h(x,y) = y^2i + (2xy-e^2y)j is conservative or not. If it is…

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Q: Determine if the vector field F=(yz,xz,xy) is conservative?

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Q: Compute the curl of the vector field F = sin(x) i+ cos(y) j+ 8z3 k

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Q: Compute the curl, V x F, of the vector field. F(x, y, z) = 7xi + 9yj + 2zk curl(F) = -7xi – 5yj +…

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Q: Determine whether F(x, y) = xi +9yj is a conservative vector field. If so, find a potential function…

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Q: 13

A: We have to find out conservative or not

Q: For the vector field F = ((y/z2+1)i + (-2x/z2+1)j + (3x2+3y2/(z2+1)2)k , calculate curlF

A: We have given vector field F=yz2+1i+1-2xz2j+3x2+3y2z2+12k=yz2+1,1-2xz2,3x2+3y2z2+12=F1,F2,F3…

Q: Determine whether the vector field is conservative. F(x, y) = 6y6(yi + 7xj) conservative not…

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Q: Determine whether the vector field…

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Q: Compute the flux of the vector field F = across sphere a? + y² + z² = 9.

A: Let F be a vector field and S be the boundary of the surface D. To find the flux of the vector…

Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Compute the curl, Vx F, of the vector field. F(x, y, z) = 5xi + 3yj + 4zk %3D curl(F) =

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Q: Calculate the line integral , F · dr, where C is the straight-line path from (2,0) to (0,2),…

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Q: Calculate the flux of the vector field F = -2i + 5æ²j – 3k, through the square of side 2 in the…

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Q: A conservative vector field is given as F = 6xe'i + (3x²e" –10yz')j-15y²z°k. Find the potential…

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Q: Find the curl of the vector field F < ya?, xz°, zyš curl F = k || + +

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Q: Compute the work done by the vector field F(x, y) = (ry, x² + y³) when moving a particle…

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Q: Find the divergence of the vector field. V(x, y, z) = 7e*Yj – 9e*Yj + 8EY²K - div(V) II

A: A detailed solution is given below.

Q: Find the work done by the vector field F = (- 3z, 4y, 2x) in moving an object along C in the…

A: Work done by any vector field F is given by ∫F.dr where r is the position vector. Dot product is the…

Q: Calculate the flux of the vector field F = x²i+ zk across the portion of the plane described as z =…

A: According to our company policy, we are supposed to answer only first question. For the rest please…

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: A=(2x-3y)ax+(2xy-y2)ay Prove the Stokes theorem for the triangular linear path in the shape that is…

A: The general form of a vector valued function in a plane is given by A=Axax+Ayay, where Ax and Ay are…

Q: xi + yj F(x, y) = k- x² + y2'

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Q: Calculate the flux of the vector field F 6i + 2x2j- 2k, through the square of side 4 in the plane y…

A: Given vector field is F→=6i→+2x2j→−2k→ We have to find the flux of the given vector field. The flux…

Q: Given the vector field F(x, y, z) = (2z3 – 4y2 + axy,x² - 8xy – bz, 6xz? –- ay + 4), compute the…

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Q: Find the divergence of the vector field F = < yx², xz", zy® div F II

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Q: Compute the flux of the vector field F = 2xi + yj through the cylindrical 5 and b = 4 and it is…

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

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Q: Compute the flux of the vector field F = 2xi + 2yj through the surface S, which is the part of the…

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Q: Determine whether the vector field…

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Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

Q: Consider the vector Field 7 and the curve C below. F(x,y) = (3 +4 ху?) 1 + 4x²yј, C is the arc of…

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Q: Find the divergence of the vector field F. F(x, y, 2) = In(7x2+ 6y? )i + 42xyj + In(6y2. + 72² )«…

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Q: Sketch the vector field F(x, y) = 3i + 5j and calculate the flux then the circulation of F| along…

A: (1) The equation of line segment is: x+3-2+3=y-1-5-1=tx+3=t⇒x=t-3y-1=-6t⇒y=-6t+1 The curve…

Q: Compute the flux of the vector field F = 2xi + 3yj through the cylindrical surface shown below, if a…

A: We have to compute the flux of the  given vector field.