Question

In each of the following, f(x,y,z) is a function defined on all of R³ with continuous second order partial derivatives, and f(x,y,z) is a vector field defined on all of R³ whose coordinate functions have continuous second order partial derivatives.

(a) True or False: If div(F)=0 then F is conservative.
(b) True or False: .
(c) True or False: If grad(f)=F then curl(F)=vec(0).
(d) True or False: div(grad(f))=0.

Answer 1

a) False. The divergence of a vector field being zero does not guarantee that the vector field is conservative. c) False. If the gradient of a function f equals a vector field F, it does not necessarily imply that the curl of F is the zero vector. d) True. The divergence of the gradient of a function f is always zero.

Step-by-step explanation:

(a) True or False: If div(F)=0 then F is conservative.

False. The divergence of a vector field being zero does not guarantee that the vector field is conservative. For a vector field to be conservative, its curl must also be zero.

(b) True or False:

Cannot answer, incomplete question.

(c) True or False: If grad(f)=F then curl(F)=vec(0).

False. If the gradient of a function f equals a vector field F, it does not necessarily imply that the curl of F is the zero vector. The curl of F can be nonzero.

(d) True or False: div(grad(f))=0.

True. The divergence of the gradient of a function f is always zero.