A vector field is given in Cartesian components by F = xy

PHYSICS · Multi-part · 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

A vector field is given in Cartesian components byF=xy x^+yz y^+zx z^.\mathbf{F} = xy\,\hat{\mathbf{x}} + yz\,\hat{\mathbf{y}} + zx\,\hat{\mathbf{z}}. Here xx, yy and zz are pure numbers and F\mathbf{F} is dimensionless; the expression is smooth everywhere. ((a)) Find ∇⋅F\nabla\cdot\mathbf{F} at the point (x,y,z)=(2, 1, 3)(x,y,z) = (2,\,1,\,3). ((b)) Find the three components of ∇×F\nabla\times\mathbf{F} at the same point.

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