University:
Yeshiva University
Course:
1412 | Calculus I
Academic year:

2020
Views:

85

Pages:

28
Author:

Dustin L

2. Let F(x, y, z) = \langlex ^ - 1 * z, y ^ - 1 * z, ln(xy)\rangle (a) Verify that F= Vf, where f(x, y, z) = z * ln(xy) (b) Evaluate F-dr, where r(t) = \langlee ^ t, e ^ (2t), r ^ 2\rangle for 1 <= t <= 3 (c) Evaluate F-dr for any path C from P = (1/2, 4, 2) Q = (2, 2, 3) contained in the region x > 0, y > 0 (d) Why is it necessary to specify that the path lies in the region where x and y are positive? SOLUTION (a) partial f partial x = z * 1/(xy) * y = x ^ - 1 * z = F_{1} partial f partial y = z * 1/(xy) * x = y ^ - 1 * z = F_{2} partial f partial z = ln(xy) = F_{3} b) r(1) = (e, e ^ 2, 1) while r(3) = (c ^ 3, c ^ 6, 9) Therefore, ( int C F* dr= f(e ^ 3, e ^ 6, 9) - f(e, e ^ 2, 1) = 9 * ln(e ^ 3 * e ^ 6) - ln(e * e ^ 2) = 78 (c) int C F* dr= f(2, 2, 3) - f(1/2, 4, 2) = 3 * ln(2 * 2) - 2 * ln(1/2 * 4) = 4 * ln(2) aligned omega r 2 partial z = omega partial z (- cos y) =0 Rightarrow partial F 2 partial z = partial F 3 partial y aligned partial F 3 partial y = partial partial y (c ^ x) = 0 Fa aligned overline partial x = overline partial x (c) = c \\ partial F 1 partial F 1 = partial partial x = partial partial x = partial F 1 partial z = partial F 1 partial z aligned partial F 1 partial z = partial partial z (e ^ x * (z + 1)) = e ^ x F satisfies the cross-partials condition everywhere. Hence, F is conservative. If f(x, y, z) a potential func-tion, then it satisfies f(x, y, z) = integrate e ^ x * (z + 1) dx = e ^ x * (z + 1) + f(y, z) = z * e ^ x + e ^ x + f(y, z) f(x, y, z) = integrate (- cos y) dy = - sin y + g(x, z) f(x, y, z) = integrate e ^ x dz = z * e ^ x + h(x, y) These three ways of writing f(x, y, z) must be equal, so z * x ^ x + e ^ x + f(y, z) = - sin y + g(x, z) = z * x ^ x + h(x, y) If we choose f(y, z) = - sin y g(x,y)=zee, and h(x, y) = e ^ x - sin y the equalities hold, so that a potential function is f(x, y, z) = z * c ^ x + e ^ x - sin y + C = (z + 1) * e ^ x - sin y + C for any constant C.

Solutions to HW 11 Math

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