Introduction to Electrodynamics · Problem 1.14

Vector Analysis — David J. Griffiths

Physics · gradient

Problem

Suppose that

f

is a function of two variables (

y

and

z

) only. Show that the gradient

\nabla f = (\partial f/\partial y)\hat{\mathbf{y}} + (\partial f/\partial z)\hat{\mathbf{z}}

transforms as a vector under rotations, Eq. 1.29. [*Hint:*

(\partial f/\partial \bar{y}) = (\partial f/\partial y)(\partial y/\partial \bar{y}) + (\partial f/\partial z)(\partial z/\partial \bar{y})

, and the analogous formula for

\partial f/\partial \bar{z}

. We know that

\bar{y} = y \cos\phi + z \sin\phi

and

\bar{z} = -y \sin\phi + z \cos\phi

; "solve" these equations for

y

and

z

(as functions of

\bar{y}

and

\bar{z}

), and compute the needed derivatives

\partial y/\partial \bar{y}

\partial z/\partial \bar{y}

, etc.]