Introduction to Electrodynamics · Problem 1.50

Vector Analysis — David J. Griffiths

Physics · vector-potentials

Problem

(a) Let

\mathbf{F}_1 = x^2\,\hat{\mathbf{z}}

and

\mathbf{F}_2 = x\,\hat{\mathbf{x}} + y\,\hat{\mathbf{y}} + z\,\hat{\mathbf{z}}

. Calculate the divergence and curl of

\mathbf{F}_1

and

\mathbf{F}_2

. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. (b) Show that

\mathbf{F}_3 = yz\,\hat{\mathbf{x}} + zx\,\hat{\mathbf{y}} + xy\,\hat{\mathbf{z}}

can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

Related theory

Vector And Scalar Potentials

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