In this problem treat ,

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

In this problem treat

x

y

z

and the field components as pure numbers, so no units need be carried. (a) For a general smooth field

\mathbf{F}

, write

\nabla\cdot(\nabla\times\mathbf{F})

out in full in Cartesian components (six terms) and show that it vanishes, naming the property of second derivatives you use. (b) Check the result on the field

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

: compute the three components of

\nabla\times\mathbf{F}

, then compute the divergence of that curl, showing how the six terms cancel in pairs. (c) Explain in two sentences how this same result follows from the fact that the flux of a curl is the same through every surface with the same rim.

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