Unit 2: Tensor Algebra, Operations & The Quotient Law

1. General Definition of a Tensor of Type $(r, s)$

In modern differential geometry and multilinear algebra, a tensor is an invariant geometric object whose components with respect to a coordinate basis transform multilinearly under changes of coordinates.

Definition 2.1 (Tensor of Rank $(r, s)$): A geometric entity is called a tensor of contravariant rank $r$ and covariant rank $s$ (or simply a tensor of type $(r, s)$ and total rank $r + s$) at point $P \in V_n$ if it is characterized by $n^{r+s}$ real numbers:

in coordinate system $x^k$, such that under an arbitrary regular coordinate transformation $x^k \to \bar{x}^k$, its components in the new system $\bar{T}$ are given by:

Special Cases of Tensor Ranks:

1. Rank $(0, 0)$: A scalar invariant $\Phi$ ($\bar{\Phi} = \Phi$, 1 component).

2. Rank $(1, 0)$: A contravariant vector $A^i$ ($\bar{A}^i = \frac{\partial \bar{x}^i}{\partial x^p} A^p$, $n$ components).

3. Rank $(0, 1)$: A covariant vector (covector / 1-form) $B_j$ ($\bar{B}_j = \frac{\partial x^q}{\partial \bar{x}^j} B_q$, $n$ components).

4. Rank $(2, 0)$: A contravariant tensor of second rank $T^{ij}$ ($\bar{T}^{ij} = \frac{\partial \bar{x}^i}{\partial x^p} \frac{\partial \bar{x}^j}{\partial x^q} T^{pq}$, $n^2$ components).

5. Rank $(0, 2)$: A covariant tensor of second rank $T_{ij}$ ($\bar{T}_{ij} = \frac{\partial x^p}{\partial \bar{x}^i} \frac{\partial x^q}{\partial \bar{x}^j} T_{pq}$, $n^2$ components).

6. Rank $(1, 1)$: A mixed tensor of second rank $T^i_{\; j}$ ($\bar{T}^i_{\; j} = \frac{\partial \bar{x}^i}{\partial x^p} \frac{\partial x^q}{\partial \bar{x}^j} T^p_{\; q}$, $n^2$ components).

2. Group Property of Tensor Transformations

The set of all admissible coordinate transformations forms a mathematical Lie group under composition. If coordinate transition $x \to \bar{x}$ has Jacobian $J_1$ and $\bar{x} \to \bar{\bar{x}}$ has Jacobian $J_2$, then the composite transformation $x \to \bar{\bar{x}}$ satisfies:

Applying this identity to the tensor transformation law verifies that the composition of two tensor transformations is itself a tensor transformation of the exact same type. Thus, tensor character is an intrinsic, coordinate-independent mathematical property.

1. Addition and Linear Combinations

Theorem 2.1 (Addition of Tensors): If $A^{i_1 \dots i_r}_{j_1 \dots j_s}$ and $B^{i_1 \dots i_r}_{j_1 \dots j_s}$ are two tensors of the exact same type $(r, s)$, then their linear combination:

is also a tensor of type $(r, s)$.

Proof: Follows directly from the linearity of the Jacobian transformation matrices. (Note: Tensors of different ranks or different index heights cannot be added, as their transformation laws do not factor).

2. Outer Product (Tensor Product $\otimes$)

Theorem 2.2 (Outer Product Theorem): Let $A$ be a tensor of type $(r, s)$ and $B$ be a tensor of type $(p, q)$. The outer product $C = A \otimes B$, defined by:

is a tensor of contravariant rank $r + p$ and covariant rank $s + q$ (type $(r+p, s+q)$).

Proof: Transform each tensor factor independently:

The transformation law factors into the product of $r+p$ contravariant Jacobian factors and $s+q$ covariant Jacobian factors. Thus $C$ is a tensor of type $(r+p, s+q)$. $\blacksquare$

3. Contraction of Indices

Contraction consists of setting one upper index equal to one lower index and summing over their common values according to the Einstein summation convention.

Theorem 2.3 (Contraction Theorem): If $T^{i_1 \dots i_r}_{j_1 \dots j_s}$ is a tensor of type $(r, s)$ with $r \ge 1$ and $s \ge 1$, then contracting one contravariant index with one covariant index (e.g., setting $i_1 = j_1 = k$ and summing):

produces a tensor of type $(r-1, s-1)$.

Proof: Transform the contracted tensor:

Observe the contracted Jacobian factors:

Substituting $\delta^q_p$ collapses the indices $p$ and $q$:

The remaining Jacobian factors are precisely those of a tensor of type $(r-1, s-1)$. $\blacksquare$

1. Inner Product of Tensors

The inner product of two tensors is defined as the outer product followed immediately by one or more contractions.

• Example 1: Vector $A^i$ and covector $B_j$. Outer product: $T^i_{\; j} = A^i B_j$ (rank 2). Contracting $i = j$ yields the scalar inner product $A^i B_i$ (rank 0).
• Example 2: Rank-2 tensor $T_{ij}$ and contravariant vector $v^j$. The inner product $w_i = T_{ij} v^j$ is a covariant vector (rank 1).
• Example 3: Two rank-2 tensors $A^{ij}$ and $B_{jk}$. Inner product $C^i_{\; k} = A^{ij} B_{jk}$ is a mixed rank-2 tensor.

2. The Trace Invariant of Mixed Tensors

For a mixed tensor of second rank $T^i_{\; j}$, contracting the upper index with the lower index yields:

By Theorem 2.3, contracting type $(1, 1)$ yields type $(0, 0)$, which is an absolute scalar invariant. In contrast, the diagonal sum of a purely covariant tensor $\sum_i T_{ii}$ is NOT an invariant under general coordinate transformations.

3. The Generalized Kronecker Delta

Definition 2.2 (Generalized Kronecker Delta): The generalized Kronecker delta of rank $2k$ is defined by the determinant:

It is $+1$ if $(j_1, \dots, j_k)$ is an even permutation of distinct indices $(i_1, \dots, i_k)$, $-1$ if an odd permutation, and $0$ if any index is repeated. It is an isotropic mixed tensor of type $(k, k)$.

1. Formulation of the Quotient Law

Testing whether a candidate array of $n^k$ functions forms a tensor by verifying the multilinear transformation law directly can be algebraically prohibitive. The Quotient Law of Tensors provides an indispensable criterion: if the inner product of a candidate entity with an arbitrary known tensor always produces a tensor, then the candidate entity is itself a tensor.

Theorem 2.4 (The Quotient Law for Rank-2 Tensors): Let $A(i, j)$ be a set of $n^2$ quantities defined in every coordinate system. If for every arbitrary contravariant vector $u^i$ and every arbitrary contravariant vector $v^j$, the scalar contraction:

is an invariant scalar ($\bar{\Phi} = \Phi$), then $A(i, j)$ are the components of a covariant tensor of rank 2 ($A(i, j) = A_{ij}$).

Proof:

1. In coordinate system $x^k$: $\Phi = A(p, q) u^p v^q$.
2. In coordinate system $\bar{x}^k$: $\bar{\Phi} = \bar{A}(i, j) \bar{u}^i \bar{v}^j$.
3. Since $u^p$ and $v^q$ are known contravariant vectors:

1. Substitute these into $\bar{\Phi}$:

1. Since $\bar{\Phi} = \Phi$, we equate both expressions:

1. Crucially, the vectors $u^p$ and $v^q$ are completely arbitrary. Setting $u^p = \delta^p_a$ and $v^q = \delta^q_b$ selects the individual $(a, b)$-component:

1. Multiplying both sides by $\frac{\partial x^a}{\partial \bar{x}^k} \frac{\partial x^b}{\partial \bar{x}^l}$ and contracting over $a$ and $b$:

This is precisely the transformation law of a covariant tensor of rank 2. $\blacksquare$

1. Symmetric and Antisymmetric Tensors

Let $T_{ij}$ be a second-rank covariant tensor:

• Symmetric Tensor: $T_{ij} = T_{ji}$ for all $i, j$.
• Antisymmetric (Skew-Symmetric) Tensor: $T_{ij} = -T_{ji}$ for all $i, j$ (implying $T_{ii} = 0$ along the diagonal).

Theorem 2.5 (Preservation of Symmetry under Coordinate Changes): If a tensor is symmetric (or antisymmetric) with respect to two indices in one coordinate system, it is symmetric (or antisymmetric) in all coordinate systems.

Proof for Symmetry: Assume $T_{pq} = T_{qp}$ in coordinates $x^k$.

Renaming dummy indices $p \leftrightarrow q$:

Thus symmetry is a true coordinate invariant. $\blacksquare$

2. Decomposition of a Rank-2 Tensor

Any second-rank tensor $T_{ij}$ can be uniquely decomposed into the sum of a symmetric tensor $S_{ij} = T_{(ij)}$ and an antisymmetric tensor $A_{ij} = T_{[ij]}$:

where:

• Number of independent components of symmetric tensor in $n$ dimensions: $\frac{n(n + 1)}{2}$.
• Number of independent components of antisymmetric tensor in $n$ dimensions: $\frac{n(n - 1)}{2}$.
• Total sum: $\frac{n(n+1)}{2} + \frac{n(n-1)}{2} = n^2$.

3. The Levi-Civita Permutation Symbol and Pseudo-Tensors

The Levi-Civita permutation symbol $\epsilon_{ijk}$ in 3 dimensions is defined by:

Under coordinate transformations:

The appearance of the Jacobian factor $\frac{1}{J}$ indicates that $\epsilon_{ijk}$ is a tensor density (or relative pseudo-tensor) of weight $-1$. To make it a true absolute tensor, it must be multiplied by $\sqrt{g}$.