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.

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A: We find the required integrtion using fundamental theorem of line Integral.

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

A: Topic- vector integration

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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 .

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Q: Compute the curl of F = (2xy, x²y + x²).

A: Here we have to find the curl of the given vector field F.

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Q: Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.

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