Let f be a scalar field and F a vector field State whether e

Let f be a scalar field and F a vector field. State whether each expression is meaningful. If so, state whether it is a scalar field or a vector field. curl f grad f div F curl(grad f) grad F grad(div F) div(grad f) grad(div f) curl(curl F) div(div F) (grad f) times (div F) div(curl(grad f))

Solution

a. curl f False, curl can be taken of only vector field

b. grad f   Vector field, gradient results in vector field

c. div F Scalar field, divergence results in scalar field

d. curl(grad f) Vector field, curl of a vector field results in vector field

e. grad F False, gradient can be taken of only scalar field

f. grad(div F)   Vector field, gradient of scalar field is a vector field

g. div(grad f)   Scalar field, divergence of vector field is a scalar field

h. grad(div f)   False, divergence of scalar field cannot be taken

i. curl(curl F)   Vector field, the curl of a vector field is a vector

j. div(div F)   False, divergence of scalar field cannot be taken

k. (grad f) ×(div F)   False, a vector and scalar field cannot be crossed

l. div(curl(grad f))   Scalar field, divergence of a vector field is a scalar field