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### Introduction to Electrodynamics

Book edition 4th edition
Author(s) David J. Griffiths
Pages 613 pages
ISBN 9780321856562

# Suppose that f is a function of two variables (y and z) only. Show that the gradient transforms as a vector under rotations, Eq 1.29. [Hint: and the analogous formula for . We know that and ”solve” these equations for y and z (as functions of and (as functions of and ), and compute the needed derivatives , etc]

The matrix proves that’s is used to transform the vector under the rotation.

See the step by step solution

## Step 1: Write the expression for the coordinates.

Consider the change in coordinates is .

Here, the variables of the function are y and z. The shifted coordinates are and are the changed coordinates. The angle of rotation is .

Rewrite the equations as,

Add the equation for the two coordinates as,

Rewrite the coordinates as,

Subtract the two equations as,

Determine the partial derivatives of the equation (1).

Determine the partial derivatives of the equation (2).

## Step 2: Determine the proof that  transform as the vector under rotation.

Write the expression for the gradient of the function with respect to y.

Write the equation for the gradient in terms of the partial derivative as,

Write the expression for the gradient in terms of the z.

Write the expression for the gradient in the matrix form.

Thus, the matrix proves that’s the is used to transform the vector under the rotation.