Elements of Field Theory

Abstract

Elementary concepts of vector-field theory are introduced and the integral theorems of Gauss and Stokes are stated. The properties of irrotational, solenoidal, and conservative fields are examined.

Questions

• 1. Show that the grad is a linear operator: $\overrightarrow{\nabla}\;\left(f+g\right)=\overrightarrow{\nabla}f+\overrightarrow{\nabla}g$, for any functions f (x,y,z) and g(x,y,z). Also show that $\overrightarrow{\nabla}\;\left(k\kern0em f\right)=k\kern0.1em \overrightarrow{\nabla}f$, where k is a constant.

• 2. Show that the grad satisfies the Leibniz rule: $\overrightarrow{\nabla}\;\left(f\kern0.1em g\right)=g\kern0.2em \overrightarrow{\nabla}f+f\kern0.2em \overrightarrow{\nabla}g$, for any functions f (x,y,z) and g(x,y,z). We say that the grad operator is a derivation on the set of all differentiable functions in R³.

• 3. Prove the vector identities (4.19) and (4.20).

• 4. Give the physical and the geometrical significance of the concepts of an irrotational and a solenoidal vector field.

• 5. (a) Show that a conservative force field is necessarily irrotational. (b) Can a time-dependent force field $\overrightarrow{F}\left(\overrightarrow{r},t\right)$ be conservative, even if it happens to be irrotational? (Hint: Is work along a given curve a uniquely defined quantity in this case? See [4]).