MATH 2E Study Guide - Midterm Guide: Parallelogram, Partial Derivative, Curve
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Problem 1 : (a) change from rectangular to spherical coordinates the point ( . 3)2 + 1 + (2 cos = z. We have and x2 + y2 + z2 = Sin (b) sketch the vector eld f(x, y) = y(cid:126)i x(cid:126)j. We have |f| =(cid:112)x2 + y2 and thus the length of the vectors are increasing the farer we are from the origin. Therefore at each point p = (x, y), f is orthogonal to r. we then get the following drawing (c) find a differentiable function : r r such that the vector eld de ned by. 2xy (1 + x2) (cid:126)i + (x)(cid:126)j, (cid:90) F d(cid:126)r, where c is the ellipse x2 = We want f to be conservative, that is and. Thus we choose (x) = ln(1 + x2). C is a closed curve, so its nal and initial points coincide.
