tensor densityTensorDensity2005-01-01 19:52:522005-02-18 00:36:34Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands here\subsection{Heuristic definition}
A tensor density is a quantity whose transformation law under change of basis involves the determinant of the transformation matrix (as opposed to a tensor, whose transformation law does not involve the determinant).
\subsection{Linear Theory}
For any real number $p$, we may define a representation $\rho_p$ of the group $GL(\mathbb{R}^k)$ on the vector space of tensor arrays of rank $m,n$ as follows:
$$(\rho_p (M) T)^{i_1, \ldots, i_n}_{j_1, \ldots j_m} = (\mathop{\rm det}(M))^p M^{i_1}_{l_1} \cdots M^{i_n}_{l_n} (M^{-1})_{k_1}^{j_1} \cdots (M^{-1})_{k_m}^{j_m} T^{i_1, \ldots, i_n}_{j_1, \ldots j_m}$$
A \emph{tensor density} $T$ of rank $m,n$ and weight $p$ is an element of the vector space on which this representation acts.
Note that if the weight equals zero, the concept of tensor density reduces to that of a tensor.
\subsection{Examples}
The simplest example of such a quantity is a scalar density. Under a change of basis $y^i = M^i_j x^j$, a scalar density transforms as follows:
$$\rho_p (S) = (\mathop{\rm det}(M))^p S$$
An important example of a tensor density is the Levi-Civita permutation symbol. It is a density of weight $1$ because, under a change of coordinates,
$$(\rho_1 \epsilon)_{j_1, \ldots j_m} = (\mathop{\rm det}(M)) (M^{-1})_{k_1}^{j_1} \cdots (M^{-1})_{k_m}^{j_m} \epsilon^{i_1, \ldots, i_n}_{j_1, \ldots j_m} = \epsilon_{k_1, \ldots k_m}$$
\subsection{Tensor Densities on Manifolds}
As with tensors, it is possible to define tensor density fields on manifolds. On each coordinate neighborhood, the density field is given by a tensor array of functions. When two neighborhoods overlap, the tensor arrays are related by the change of variable formula
$$T^{i_1, \ldots, i_n}_{j_1, \ldots j_m} (x) = (\mathop{\rm det}(M))^p M^{i_1}_{l_1} \cdots M^{i_n}_{l_n} (M^{-1})_{k_1}^{j_1} \cdots (M^{-1})_{k_m}^{j_m} T^{i_1, \ldots, i_n}_{j_1, \ldots j_m} (y)$$
where $M$ is the Jacobian matrix of the change of variables.