Homework Help: vector calculus

Posted by Donny on Monday, November 28, 2011 at 9:26pm.

Show that the given line integral is independent of path.Then, evaluate the line integral I by finding a potential function f and applying the Fundamental Theorem of Line Integrals. I=Бч_{(0,0)}^{(1,2)}(x+y)dx+(x-y)dy

vector calculus - Steve, Tuesday, November 29, 2011 at 12:20am
If F is a conservative field, then there will be a G such that F = ∇G.

If we let G(x,y) = 1/2 (x^2 + 2xy - y^2) then
∇G = 1/2 (2x dx + 2y dx + 2x dy - 2y dy)
= (x+y)dx + (x-y)dy

So, F is a conservative field and the line integral is path-independent.

So, just evaluate G at the limits of integration

G(1,2) = 1/2 (1+4-4) = 1/2
G(0,0) = 0

So, the integral evaluates to just 1/2

If I recall my vector calculus correctly...

Answer this Question

Related Questions

Multivariable Calculus - Green's Theorem - Compute the line integral of F...
Calculus - Use the shell method to set up, but do not evaluate, an integral ...
Math - Hi, I need urgent help with these 3 integrals problems ... been stuck on ...
Calculus 2 - The question is: Evaluate the improper integral for a>0. The...
Calculus - evaluate the integral or state that it diverges. Check if I did it ...
calculus - consider the function f(x) = e^x(sinNx) on the interval [0,1] where N...
calculus - consider the function f(x) = e^x(sinNx) on the interval [0,1] where N...
Calculus - Evaluate the integral: 16csc(x) dx from pi/2 to pi (and determine if ...
Calculus integral - evaluate the integral: integral from -pi/4 to 0 for the ...
math/calculus - You are given the four points in the plane A(-3,6),B(0,-6),C(5,...

For Further Reading

First Name:
School Subject:
Answer: