Question

Find a potential function for the vector field

Answer 1

(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
`x` gives

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

Differentiating both sides of this with respect to
`y` 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}`