16.3 Homework - The Fundamental Theorem (Homework)

8. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALCET9 16.3.023. Consider F and C below. F(x, y, z) = yze*zi + e*zj + xye*zk, C: r(t) = (+2 + 4)i + (+2 - 1)j + (+2 - 4t) k, osts4 (a) Find a function f such that F = Vf. f( x, y, Z ) = (b) Use part (a) to evaluate F . dr along the given curve C. 138 X Need Help? Read It Watch It\f9. [-/1 Points] DETAILS SCALCET9 16.3.025. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Show that the line integral is independent of path. 2xe Ydx + (2y - x e-V)dy, C is any path from (1, 0) to (5, 1) The functions 2xey and 2y - xe" have continuous first-order derivatives on R2 and - (2xe-V) = ax = ( 2y - x2e -y ), so F (x, y) = ( i + (2y - x2e-V) j is a conservative vector field by the theorem given below, hence the line integral is independent of path. Theorem: Let F = Pi + Qj be a vector field on an open simply-connected region D. Suppose that P and Q have continuous first-order partial derivatives and - ay ax OP = 2 throughout D. Then F is conservative. Evaluate the integral. Need Help? Read It Watch It10. [-/1 Points] DETAILS SCALCET9 16.3.034. Let F = Vf, where f(x, y) = sin(x - 7y). Find curves C, and C, that are not closed and satisfy the equation. (a) F . dr = 0 C: r(t) = Outs1 (b ) F - dr = 1 C 2 C,: r(t) = Osts1