MATH 210 Study Guide - Final Guide: Unit Vector, Directional Derivative, Maxima And Minima
Document Summary
Problem 1 solution: consider the vector eld f = hy2, 2xy + 2yi. (a) show that f is conservative. (b) find a potential function such that f = . (c) compute zc. F d s along any path c from ( 1, 2) to (3, 0). Solution: (a) in order for the vector eld f = hf (x, y), g(x, y)i to be conservative, it must be the case that: Using f (x, y) = y2 and g(x, y) = 2xy + 2y we get: Thus, the vector eld is conservative. (b) if f = , then it must be the case that: Using f (x, y) = y2 and integrating both sides of equation (1) with respect to x we get: We obtain the function h(y) using equation (2). Using g(x, y) = 2xy + 2y we get the equation: We now use equation (3) to obtain the left hand side of the above equation.
