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=ç_{(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

- Math - Hi, I need urgent help with these 3 integrals problems ... been stuck on ...
- Multivariable Calculus - Green's Theorem - Compute the line integral of F = <...
- Calculus - I have two questions, because I'm preparing for a math test on monday...
- calculus - a) Let f(z) = z^2 and γ(t) = 1 + it^3, t ∈ [0,1]. i) Write...
- Calculus - for the function F(t)=ln(t^2), let f(t)=F'(t). Write the integral ...
- Calculus - "Leave the answer as a definite integral, but indicate how it could ...
- Math - Evaluate the line integral SC F · dr, where C is given by the vector ...
- math - Use Green's Theorem to evaluate the line integral along the given ...
- Calculus - use te fundamental theorem of calculus to evaluate the integral ...
- Calculus - Use the Fundamental Theorem to evaluate the integral from 0 to 3 of ...

More Related Questions