Consider the vector field F(x,y,z) = (3x²y)i + (x³ + 1) + (92²) and the curve C made up of the quarter circle cut from the sphere x² + y² + z² = 25 by the plane x = 3, from (3,4,0) to (3,0,4) traveling counterclockwise when looking from the positive x -axis. a. Use a line integral (as in Section 15.2) to compute the work done by the vector field in moving a particle along C. b. Show that the vector field is conservative, then compute the work done by the vector field in moving a particle along C using a potential function (as in Section 15.3) for the vector field.

Q: Sketch the vector field in the plane along with its horizontal and vertical components at a…

A: We shall use MATLAB to plot the given vector field on the circle.  MATLAB CODE F = @(x,y)…

Q: ector product of two vector fields: V × (F x G) = (V · G)F – (V · F)G + (G · V)F – (F · V)G %3D

A: Part iii Given, The identity of the vector product of two vector fields is proved.

Q: Consider the vector field v(x)

A: Given, the vector field        v(x)=yx2+y2i^-xx2+y2j^+6k^                               ...(1)

Q: Show that Gauss's Theorem holds for the vector field F(x, y)=(xy2, -yx2) in a side square 3.

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Q: Find the k-component of (curl F) for the following vector field on the plane. F = (9xe )i + (ye*)j…

A: The given problem is to find the k-component of the curl of the given vector field F=(9xey)i +…

Q: he vector v = , is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1).  Find a.

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Q: Give a vector parametric equation for the line that passes through the point (3, 5, –5), parallel to…

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Q: Let F=(9xy,7y,8z)=(9xy,7y,8z). The curl of F=( Is there a function f such that F=∇f=∇?  (y/n)

A: The given function .We have to find the .

Q: (b) part of parabola y = −2x² from (−1, −2) to (2, —–8)

A: Given Equation of parabola y=-2x2 from -1, -2 to 2, -8.

Q: Evaluate the 1-form φ = x2yz dx − z dy on the vector field V (x, y, z) = xU1(x, y, z) − yzU2(x, y,…

A: We need to evaluate the 1-form of  ϕ=x2yz dx - z dy.....(1) in the vector field  V(x,y,z) = x…

Q: For any vector field F, show that div(curl F) = 0.

A: We need to prove that for any vector field F,  div(curl F) = 0.

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

A: Denote the unit sphere by S If the sphere is closed,  then the flux of v(x, y, z) out of S is given…

Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

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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: the vector fieldR-(x² 2 2 what is 282

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Q: Consider the following function. T: R? - R, T(x, y) = (2x?, 3xy, y²) Find the following images for…

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Q: Consider the square C oriented counter clockwise, with vertices (0, 0), (2, 0), (0, 2), (2, 2) and 7…

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Q: 2. A vector field is specified as G = 24xy ā, + 12(x² + 2)ā, + 18z²ā,. Given two points, P(1, 2, -1)…

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Q: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

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Q: 3. Find a parametrization for (a vector valued function that traces) the line which passes through P…

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Q: 285. Find the scalar equation of the plane that passes through point P(-1, 2, 1) and is…

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Q: Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position vector of a point…

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Q: Compute the curl of F = (3y2, ay, 22).

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Q: Find curl(zi + xj+ yk).

A: Using definition of curl of vector.