MTH3008 Lecture 13

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Last time we wrapped up Chapter 5 - tensors in generalised coordinate systems - covering covariant, contravariant, and mixed components of second-rank tensors, associated tensors (raising and lowering indices via the metric tensor), and symmetry/antisymmetry. We now open Chapter 6: Tensor Algebra, which asks a natural follow-up question - how do we build new tensors from existing ones?

Products of Tensors So Far

Before defining new operations, recall the tensor-related products we already know:

• The scalar product $\mathbb{R}\times {\mathbb{R}}^3\to {\mathbb{R}}^3$: $(r,\mathbf{v})\mapsto r\mathbf{v}=(r{v}_1,r{v}_2,r{v}_3)$.
• The Kronecker Delta ${\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$: $(\mathbf{v},\mathbf{u})\mapsto \mathbf{v}\cdot \mathbf{u}=v_1{u}_1+v_2{u}_2+v_3{u}_3$.
• The Alternating Tensor ${\mathbb{R}}^3\times {\mathbb{R}}^3\to {\mathbb{R}}^3$: $(\mathbf{v},\mathbf{u})\mapsto \mathbf{v}\times \mathbf{u}$.
• The matrix product ${\text{Mat}}_{m\times k}(\mathbb{R})\times {\text{Mat}}_{k\times n}(\mathbb{R})\to {\text{Mat}}_{m\times n}(\mathbb{R})$: $(A,B)\mapsto AB$.

Each takes tensors of certain ranks and produces a tensor of (possibly different) rank. Chapter 6 formalises the general machinery behind this.

The operations we cover in this chapter are:

1. Addition and subtraction of tensors,
2. Outer product,
3. Contraction of tensors.

This lecture focuses on the first of these.

Addition of Second-Rank Tensors

Definition and Proof for Covariant Components

Let $A_{ik}$ and $B_{ik}$ be the covariant components of two second-rank tensors. Define their tensor addition componentwise:

$$C_{ik}=A_{ik}+B_{ik}.$$

For this to be meaningful, $C_{ik}$ must itself be a covariant second-rank tensor - that is, it must obey the correct transformation law. Let us verify this.

Since $A_{ik}$ and $B_{ik}$ are covariant second-rank tensors, they transform as:

$$A_{ik}^{\prime }=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{A}_{mn},B_{ik}^{\prime }=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{B}_{mn}.$$

Now compute $C_{ik}^{\prime }$:

$$C_{ik}^{\prime }=A_{ik}^{\prime }+B_{ik}^{\prime }=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{A}_{mn}+L_{i^{\prime }}^m{L}_{k^{\prime }}^n{B}_{mn}=L_{i^{\prime }}^m{L}_{k^{\prime }}^n(A_{mn}+B_{mn})=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{C}_{mn}.$$

This is exactly the covariant second-rank transformation law. So $C_{ik}$ is indeed a covariant second-rank tensor.

Important

Adding two covariant second-rank tensors $A_{ik}$ and $B_{ik}$ produces another covariant second-rank tensor: $\boxed{A_{ik}+B_{ik}=C_{ik}}$

Extension to Other Index Structures

The same argument applies to any matching index structure. By identical reasoning:

• Two contravariant second-rank tensors sum to a contravariant second-rank tensor:

$$\boxed{A^{ik}+B^{ik}=C^{ik}}$$

• Two mixed second-rank tensors (of the same type) sum to a mixed second-rank tensor:

$$\boxed{A_{\cdot k}^i+B_{\cdot k}^i=C_{\cdot k}^i}$$ $$\boxed{A_i^{\cdot k}+B_i^{\cdot k}=C_i^{\cdot k}}$$

The proof is the same each time: factor the common transformation matrices out of the sum.

Addition of Tensors with Different Index Positions

What happens if we try to add tensors with different index structures - say a covariant tensor $A_{ik}$ and a contravariant tensor $B^{ik}$?

Their transformation rules are fundamentally different:

$$A_{ik}^{\prime }=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{A}_{mn},B^{\prime ik}=L_m^{i^{\prime }}{L}_n^{k^{\prime }}{B}^{mn}.$$

For the sum to be a tensor, it would need to satisfy one of the four second-rank transformation laws (covariant, contravariant, or one of the two mixed types). But:

$$A_{ik}^{\prime }+B^{\prime ik}=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{A}_{mn}+L_m^{i^{\prime }}{L}_n^{k^{\prime }}{B}^{mn}.$$

The two terms involve different transformation matrices ($L_{i^{\prime }}^m$ vs $L_m^{i^{\prime }}$), so there is no common factor to extract. This expression does not satisfy any of the four transformation rules.

Warning

Tensors with different index structures cannot be added. The expression $A_{ik}+B^{ik}$ is not a tensor. Both operands must share the same rank and the same covariant/contravariant structure.

Addition of Arbitrary-Rank Tensors

Everything above generalises to tensors of any rank. We can add any number of tensors provided they share the same rank and index structure:

$$P_{ijk}=A_{ijk}+B_{ijk}+C_{ijk}$$ $$Q^{ijk}=D^{ijk}+E^{ijk}$$ $$R_i^{\cdot jk\ell }=F_i^{\cdot jk\ell }+G_i^{\cdot jk\ell }$$

The proof is always the same: each summand transforms with identical matrices, so those matrices factor out of the sum.

Important

Tensors of different ranks or different structures (covariant, contravariant, or different types of mixed) cannot be added.

Worked Example - Matrix Addition

Example

Recall that rank-two tensors can be represented as matrices. Let $M_{ij}=\left(\begin{array}{ccc}0& -2& -3\\ 1& 2& 5\\ 1& 3& 4\end{array}\right),N_{ij}=\left(\begin{array}{ccc}7& 8& 9\\ 6& 5& 4\\ 1& 2& 3\end{array}\right).$ Their sum $C_{ij}=M_{ij}+N_{ij}$ has $(i,j)$-component $M_{ij}+N_{ij}$, giving $[C_{ij}]=\left(\begin{array}{ccc}7& 6& 6\\ 7& 7& 9\\ 2& 5& 7\end{array}\right).$

This is just ordinary matrix addition - nothing surprising. The point is that the tensorial character (the transformation law) is preserved automatically.

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Pre-Lecture Notes from mth3008 lecture notes 13.pdf

• Chapter 6: Tensor Algebra covers three operations for building new tensors: addition, outer product, and contraction.
• Known products (scalar, dot, cross, matrix) are all special cases of tensor products of various ranks.
• Addition of tensors is defined componentwise: $C_{ik}=A_{ik}+B_{ik}$.
• Proof that the sum is a tensor: factor common transformation matrices $L_{i^{\prime }}^m{L}_{k^{\prime }}^n$ out of the sum.
• Works identically for contravariant ($A^{ik}+B^{ik}$) and mixed ($A_{\cdot k}^i+B_{\cdot k}^i$) components.
• Cannot add tensors of different rank or index structure - the transformation matrices differ and cannot be factored.
• Extends to arbitrary rank: $P_{ijk}=A_{ijk}+B_{ijk}+C_{ijk}$, etc.
• Matrix example: adding two $3\times 3$ covariant tensors is ordinary matrix addition.
• Next lecture: outer product and contraction of tensors (continuing Chapter 6).