Question

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

Answer 1

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...