1. Answer the following questions involving the potential function 0=r = 4 - x2 - y 2 - -22
a. What is the mathematical equation describing the associated vector field?
b. Illustrate with a sketch showing direction and approximate relative magnitude
some representative vectors in the field in the x-y plane by setting Z = 0.
C. Is this vector field conservative? Why, or Why not?
Based on the picture at right, describe in words
whether you think the circulation around the curve
shown is positive, negative or zero.
b. Compute the circulation for the following:
=(x-y,x), - C:r(f)=(2cost,2sin/) = for Ost<2rt
C. Is the vector field in b. the same as that plotted in the
figure at right? Explain your answer.
3. Use a line integral to compute the work required to move an object via a straight
P(0,0) to Q(2,5) using the variable force F = (x2,x).
4. Use Green's Theorem to evaluate both the circulation and flux for the vector field F=(y²,2x²
along the curve C, defined as a triangle with vertices at (0, 0), (1, 0), (0, 2) and assuming
counter-clockwise positive. Sketch the vector field and its boundary curve to help you
visualize the problem.
5. Use a surface integral along with variable transformations x=rcos©, y=rsin 0 to compute
area of that portion of the surface described by the equation Z = x2 + y 2 which
plane Z = 1. Sketch the surface in 3D and its projection onto the r-0plane.
6. Use the divergence theorem to compute the net outward flux of the field F = (x.,2y7,3z
across the closed surface bounded from above by the surface defined by x2 + y2
and its circular "bottom", which lies in the x-y plane.
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