Vector Calculus: Operator Questions Answered

[Savant13]

I'm looking into vector calculus right now, and I have a few questions.

* is the dot operator

What is the difference between \nabla * F and F * \nabla ?

What is \nabla ^2 F, where F is a vector field?

 

[dx]

∇⋅F is ∂(F_x)/∂x + ∂(F_y)/∂y + ∂(F_z)/∂z which is a scalar function. F⋅∇ is (F_x)(∂/∂x) + (F_y)(∂/∂y) + (F_z)(∂/∂z) which is an operator.

∇²F for a vector field F is just the vector field with components ∇²(F_x), ∇²(F_y) and ∇²(F_z).

 

[Savant13]

How did you get the symbols to work?

 

[ice109]

f*del can be thought of as the directional derivative in the direction of f