Question

Let f be a scalar field and F be a vector field. The following expressions either represent scalar fields, vector fields, or are completely meaningless. Determine which of the three applies to each expression and briefly explain why.
(a) curl f
(b) Δf
(c) divF
(d) curl(Δf)
(e) ΔF
(f) Δ(divF)
(g) div(Δf)
(h) Δ(div f)
(i) curl(curlF)
(j) div(divF)
(k) (Δf) x (divF)
(l) div(curl(Δf))

Δ should be upside down as for a gradient sign. Thanks for any help you give! No need for in depth explanation, but it would help if you put a couple words down for how you got an answer.

Videos

Answer 1

Answer:

(a) curl f - meaningless

(b) grad f -Δf- vector field

(c) div F - scalar field

(d) curl( grad f )curl (Δf)- vector field (e) grad F -ΔF- meaningless

(f) grad( div F )-Δ(divF) - vector field

(g) div( grad f ) - div(Δf)- scalar field

(h) grad ( div f ) -Δ(div f)-meaningless

(i) curl ( curl F ) - vector field

(j) div( div F ) - meaningless

(k) ( grad f ) x ( div F ) -(Δf) x (divF)-meaningless

(l) div( curl( grad f )) -div(curl(Δf))-scalar field

Step-by-step explanation:

(a) curl f - meaningless; a curl can only be taken of a vector field

(b) grad f - vector field; a gradient results in a vector field

(c) div F - scalar field; a divergence results in a scalar field

(d) curl( grad f ) - vector field; the curl of a vector field results in a vector field

(e) grad F - meaningless; a gradient can only be taken of a scalar field

(f) grad( div F ) -vector field ; the gradient of a scalar field is a vector field

(g) div( grad f ) - scalar field; the divergence of a vector field is a scalar field

(h) grad ( div f ) - meaningless; the divergence of a scalar field can not be taken

(i) curl ( curl F ) - vector field; the curl of a vector field is a vector field

(j) div( div F ) - meaningless; the divergence of a scalar field can not be taken

(k) ( grad f ) x ( div F ) - meaningless; a vector and scalar field cannon be crossed

(l) div( curl( grad f )) - scalar field; the divergence of a vector field is a scalar field