7. Let V =f. 0 be open in R2. A function F : V --+ R 2
Question:
7. Let V =f. 0 be open in R2. A function F : V --+ R 2 is said to be conservative on V if and only if there is a function f : V --+ R such that F = '\l f on V. Let
(x, y) E V and let F = (P, Q) : V --+ R2 be continuous on V.
(a) Suppose that C(x) is a horizontal line segment terminating at (x, y), i.e., a line segment of the form L((X1, y); (x, y)), oriented from (Xl, y) to (x, y). If C(x) is a subset of V, prove that
~ 1 F· Tds = P(x,y).
uX c(x)
Make and prove a similar statement for 0/ oy and vertical line segments in V terminating at (x, y).
(b) Let (xo, Yo) E V. Prove that
(*) fa F·Tds=O
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Related Book For
Question Posted: