what is the correct answer?

Question

Calculate the curl of F = <-y, xy, z>

A. curl F = <1, 0, y+1>

B. curl F = <0,0,y+1>

C. curl F = 0

D. curl F = 1

Answer 1

So the answer is B.

Answer 2

Another way to remember the curl is from the Levi-Civita epsilon symbol:

In this formulation, any IN ORDER permutation of (x,y,z) yields an ε^ijk value of 1:

{(x,y,z), (y,z,x), (z,x,y)} and any OUT OF ORDER permutation yields -1:

{(x,z,y), (y,x,z), (z,y,z)}

Thus, the curl is:

x-component: (x,y,z) - (x,z,y)

y-component: (y,z,x) - (y,x,z)

z-component: (z,x,y) - (z,y,x)

Where the arguments are Fⁱ, ∂/∂x^j, F^k Or:

x-component = ∂F^y/∂z - ∂F^z/∂y

y-component = ∂F^z/∂x - ∂F^x/∂z

z-component = ∂F^x/∂y - ∂F^y/∂x

Hope that's helpful, I always found that easier than doing the determinant. Just keep the x,y,z in order for the positive term and switch it for the negative term for each component.