content_image/306319.jpg content_image/306320.jpg content_image/306321.jpg content_image/306326.jpg
Therefore we find an identity we will use often in the following: 

similarly 

## F.1.2 T µν from Varying the Metric 

<img src='content_image/306319.jpg'>

<img src='content_image/306320.jpg'>

We will use variation with respect to the metric to find the (automatically) symmetric, gauge-invariant form of T µν [319]: 

<img src='content_image/306321.jpg'>

where g is the determinant of the metric and the coefficient out front gives the proper normalization for the usual stress-energy tensor (assuming a + − −− type metric). 

## E&M 

As a warmup exercise we use Eq. (F.12) to derive the usual E&M energy density from its Lagrangian, 

<img src='content_image/306326.jpg'>

