Question

For each of the following vector fields

F, decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potentialfunction f (that is, ∇f = F) with f(0,0)=0. If it is not conservative, type N.
A. F(x,y)=(−16x+2y)i+(2x+10y) j f(x,y)= _____
B. F(x,y)=−8yi−7xj f(x,y)=_____
C. F(x,y)=(−8sin y)i+(4y−8xcosy)j f(x,y)=_____

Answer 1

(A)

`(\partial f)/(\partial x)=-16x+2y`

`\implies f(x,y)=-8x^2+2xy+g(y)`

`\implies(\partial f)/(\partial y)=2x+(\mathrm dg)/(\mathrm dy)=2x+10y`

`\implies(\mathrm dg)/(\mathrm dy)=10y`

`\implies g(y)=5y^2+C`

`\implies f(x,y)=\boxed{-8x^2+2xy+5y^2+C}`

(B)

`(\partial f)/(\partial x)=-8y`

`\implies f(x,y)=-8xy+g(y)`

`\implies(\partial f)/(\partial y)=-8x+(\mathrm dg)/(\mathrm dy)=-7x`

`\implies (\mathrm dg)/(\mathrm dy)=x`

But we assume
`g(y)` is a function of
`y` alone, so there is not potential function here.

(C)

`(\partial f)/(\partial x)=-8\sin y`

`\implies f(x,y)=-8x\sin y+g(x,y)`

`\implies(\partial f)/(\partial y)=-8x\cos y+(\mathrm dg)/(\mathrm dy)=4y-8x\cos y`

`\implies(\mathrm dg)/(\mathrm dy)=4y`

`\implies g(y)=2y^2+C`

`\implies f(x,y)=\boxed{-8x\sin y+2y^2+C}`

For (A) and (C), we have
`f(0,0)=0`, which makes
`C=0` for both.