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| Cat gets Carroll |
The very first on tensor rank comes from early in the book (page 21) where we are told that: a tensor ##T## of type (or rank) ##\left(k,l\right)## is a multilinear map from a collection of dual vectors and vectors to ##\mathbf{R}##:
$$T:\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{k\ times}\times\left(T_p\times\cdots\times T_p\right)_{l\ times}\rightarrow\mathbf{R}$$
That is Carroll's 1.56 and I am pretty sure he has that the wrong way round. It should say: a tensor ##T## of type (or rank) ##\left(k,l\right)## is a multilinear map from a collection of ##k## vectors and ##l## dual vectors to ##\mathbf{R}##:$$T:\left(T_p\times\cdots\times T_p\right)_{k\ times}\times\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{l\ times}\rightarrow\mathbf{R}$$But it turns out that you don't really need to know what ##k## and ##l## are separately in General Relativity (because we always have a metric). You only need to know the total rank ##k+l##!
Contents
- What tensor rank?
- Multi-dimensional Chain Rule
- Partial derivative gives Kronecker delta: Coordinates, Vectors, Tensors
- Partial derivatives commute
- Metric is always symmetric
- Contracting with metric lowers / raises index
- You can lower or raise indices on a tensor equation
- Swap indices with metric or any similar tensor
- Inverse of a matrix
- The determinant of the inverse is reciprocal of the determinant
- Determinant of a tensor in terms of Levi-Civita symbol
- Inverse tensor
- A relationship for the derivative of the determinant
- Fully contracted symmetric × antisymmetric tensor vanishes
- Symmetrising a tensor equation
- Two formulas involving four-velocity
- Second formula
- The projection tensor on four-velocity
- Contra / co-variant tensor transformation matrices
- Tensor contractions using matrices
Details in Commentary Tensor Tricks.pdf
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