Let vector field to be F(x, y) = (-y, -x). Is F(x, y) a conservative vector fiel

Question

Let vector field to be F(x, y) = (-y, -x). Is F(x, y) a conservative vector field? Does the line integrals of this F(x, y) have path independence property? (c) Find the potential function f(x, y) of this vector field F(x, y). (d) Let the curve to be y = x^3 from (1, 1) to (2, 8). Find the line integral using the following fundamental theorem for line integrals. integral_C F middot dr = f(r(t_2)) - f(r(t_1)). (e) Find the line integral by integral_c F(x, y) middot dr = integral_a^b F (r(t)) r'(t)dt when the closed curve is (1, 0)^line (1, 1)^line (0, 0). (f) Same problem as part (e), but evaluate by one of the statements of the fundamental theorem for line integrals.

Explanation / Answer

a> F(x,y) = <-y , -x>

in the function F let P = -y and Q = -x

then for F to be consecvaive we need dP/dy to be equal to dQ/dx

now dP/dy = -1

and dQ/dx = -1

since dP/dy = dQ/dx , then we could say that F(x,y) is a conservative vector field.

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