1. Answer the following questions involving the potential function ...

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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 of 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? 2. YA a. Based on the picture at right, describe in words 4 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 x C. Is the vector field in b. the same as that plotted in the figure at right? Explain your answer. 4 3. Use a line integral to compute the work required to move an object via a straight line from 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 the area of that portion of the surface described by the equation Z = x2 + y 2 which lies below the 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 2 and its circular "bottom", which lies in the x-y plane.

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