Find a potential function for the vector field

Question

asked Jun 5, 2023 125k views

1 Answer

Aviade · Answer 1 · 2023-06-10T17:34:43+0000

(a) We want to find a scalar function
f(x,y,z) such that
$\mathbf F = \\abla f$ . This means

$(\partial f)/(\partial x) = 2xy + 24$

$(\partial f)/(\partial y) = x^2 + 16$

Looking at the first equation, integrating both sides with respect to
gives

f(x,y) = x^2y + 24x + g(y)

Differentiating both sides of this with respect to
gives

$(\partial f)/(\partial y) = x^2 + 16 = x^2 + (dg)/(dy) \implies (dg)/(dy) = 16 \implies g(y) = 16y + C$

Then the potential function is

$f(x,y) = \boxed{x^2y + 24x + 16y + C}$

(b) By the FTCoLI, we have

$\displaystyle \int_((1,1))^((-1,2)) \mathbf F \cdot d\mathbf r = f(-1,2) - f(1,1) = 10-41 = \boxed{-31}$

$\displaystyle \int_((-1,2))^((0,4)) \mathbf F \cdot d\mathbf r = f(0,4) - f(-1,2) = 64 - 41 = \boxed{23}$

$\displaystyle \int_((0,4))^((2,3)) \mathbf F \cdot d\mathbf r = f(2,3) - f(0,4) = 108 - 64 = \boxed{44}$