Find the magnitude of the curl at (1,1,2)

Question

https://i.imgur.com/GXSebel.png

Answer 1

Curl operates on a vector field, and A is such a field.

A=<xz,-y²,2x²y>. Let A₁=xz, A₂=-y², A₃=2x²y.

Curl uses the grad operator ∇=<∂/∂x,∂/∂y,∂/∂z> and the vector product:

∇᰽A=<∂A₃/∂y-∂A₂/∂z,-∂A₃/∂x+∂A₁/∂z,∂A₂/∂x-∂A₁/∂y>=<2x²,-4xy+x,0>.

Now we plug in (1,1,2):

Curl(A)=<2,-3,0>.

Magnitude |Curl(A)|=√(2²+3²+0²)=√13=3.61.