Tensors and Relativity: Chapter 3

Tensor derivatives and gradients

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The partial derivative of a tensor is not generally itself a tensor although it is for a scalar : tex2html_wrap_inline1964 is a 0/1 tensor.

For a vector  tex2html_wrap_inline1692 , we have

eqnarray1523

It follows that tex2html_wrap_inline1970 transforms like a 1/1 tensor only if the second term is zero, i.e.

eqnarray1533

where tex2html_wrap_inline1974 and tex2html_wrap_inline1976 are constants. This is true for flat space with coordinates (ct,x,y,z) under the Poincaré transformations and in this case tex2html_wrap_inline1974 maybe interpreted as a 4D rotation matrix.

  • Note however that it is not true for flat space with a general coordinate system or in the curved spacetime of General Relativity.

We can define the derivative of a general M/N tensor tex2html_wrap_inline1888 along a curve parameterized by the proper time tex2html_wrap_inline1986 as follows:

equation1543

If the basis vectors and one- forms are the same everywhere then:

equation1545

where

equation1547

and tex2html_wrap_inline1746 is the tangent to the curve. Thus tex2html_wrap_inline1990 is also like a M/N tensor, written as

equation1551

where tex2html_wrap_inline1994 means tex2html_wrap_inline1996 . We can then define a M/N+1 tensor tex2html_wrap_inline2000 :

equation1558

This is the tensor gradient  [ remember the gradient of a scalar tex2html_wrap_inline1762 ].