# Invariance of the divergence of a vector field

9. Show that the operator Del_dot (divergence) is invariant under a coordinate transformation.

(Please show complete steps)

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#### Solution Preview

The divergence of a vector field F is given by:

Div F = d/dx_i F_i (1)

where we sum over the repeated index i (Einstein summation convention). Let's consider an arbitrary orthogonal transform S from old coordinates x to new coordinates y:

y_r = S_{r,k} x_{k} (2)

F is a vector field, and it transforms according to:

F'_r = S_{r,k} F_k (3)

The divergence in the transformed frame is, by definition (1), given by

d/dy_i F'_i ...

#### Solution Summary

We show in detail why the divergence of a vector field is invariant under a coordinate transform.

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