Invariance of the divergence of a vector field
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9. Show that the operator Del_dot (divergence) is invariant under a coordinate transformation.
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Solution Summary
We show in detail why the divergence of a vector field is invariant under a coordinate transform.
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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 ...
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