?Let f be a scalar field and \(\mathbf{F}\) a vector field. State whether each

Chapter 14, Problem 14

(choose chapter or problem)

Let f be a scalar field and \(\mathbf{F}\) a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field.

(a) \(\operatorname{curl} f\)

(b) \(\operatorname{grad} f\)

(c) \(\operatorname{div} \mathbf{F}\)

(d) \(\operatorname{curl}(\operatorname{grad} f)\)

(e) \(\operatorname{grad} \mathbf{F}\)

(f) \(\operatorname{grad}(\operatorname{div} \mathbf{F})\)

(g) \(\operatorname{div}(\operatorname{grad} f)\)

(h) \(\operatorname{grad}(\operatorname{div} f)\)

(i) \(\operatorname{curl}(\operatorname{curl} \mathbf{F})\)

(j) \(\operatorname{div}(\operatorname{div} \mathbf{F})\)

(k) \((\operatorname{grad} f) \times(\operatorname{div} \mathbf{F})\)

(1) \(\operatorname{div}(\operatorname{curl}(\operatorname{grad} f))\)