MTH3008 Lecture 1

Jesus Najera, a friend of the lecturer

Tensors are mathematical objects that are invariant under a change of coordinates & have components that change in predictable ways.

Tensors are generalisations of vectors, with a rank corresponding to different quantities:

• Rank 0 tensor: scalar,
• Rank 1 tensor: vector,
• Rank 2 tensor: matrix,
• Etc. (in higher dimensions).

This will be formally defined in later lectures

Our convention will be keeping vectors three-dimensional, just for conceptual simplicity, i.e., $\mathbf{v}=(v_{1,v_2,v_3}\in {\mathbb{R}}^3)=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$ - either notation is fine, but normally we use a completely different notation…

Suffix Notation

Suffix Notation, or Index Notation, is simply when write a sum without $\Sigma$, e.g.,

$$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3=\sum _{j=1}^3{a}_j{b}_j\to \boxed{a_j{b}_j}$$

This simplifies notation considerably, especially for more complex expressions involving multiple sums, like $(\mathbf{a}\cdot \mathbf{b})(\mathbf{c}\cdot \mathbf{d})=a_j{b}_j{c}_k{d}_k$.

Types of Index

The index that appears in the sum (exactly twice for each distinctly summed term), is called a dummy index. These can be any variable without changing the result, in any order, as they don’t show up when evaluated.

Alternatively, free indices only show up once. These represent the components within the tensors, like the $i$th component of a vector: $(\lambda \mathbf{v}{)}_i=v_i$. The number of these free indices hence indicates the number of components the tensor has - its rank. Naturally, these must be consistent throughout a problem, as they compare components across multiple vectors.

Notation Conversion

We can convert from vector notation to suffix notation by simply writing out the sum. For example, $((\mathbf{a}\cdot \mathbf{c})\mathbf{b}{)}_i={\sum }_{j=1}^3(a_j{c}_j)b_i$, and then remove the sum to get suffix notation: $a_j{c}_j{b}_i$, or $a_j{b}_i{c}_j$ - again, order doesn’t matter.

For more complex conversions, we can follow a step-by-step method. For example, for a vector equation, $\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}=|\mathbf{a}{|}^2(\mathbf{b}\cdot \mathbf{v})\mathbf{a}$…

1. Introduce free index, like $i$: $(\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}{)}_i=((\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})\mathbf{a}{)}_i:|\mathbf{a}|:=\sqrt{\mathbf{a}\cdot \mathbf{a}}$;
2. Distribute free index, for each vector.: $u_i+(\mathbf{a}\cdot \mathbf{b})v_i=(\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})a_i$;
3. Introduce dummy indices, for each scalar: $\boxed{u_i+a_j{b}_j{v}_i=a_k{a}_k{b}_{\ell }{v}_{\ell }{a}_i}$.

Example (simple expression)

$(\mathbf{a}\cdot \mathbf{b})\mathbf{u}+|\mathbf{c}{|}^2\mathbf{v}=((\mathbf{a}\cdot \mathbf{b})\mathbf{u}{)}_i+((\mathbf{c}\cdot \mathbf{c})\mathbf{v}{)}_i=(\mathbf{a}\cdot \mathbf{b})u_i+(\mathbf{c}\cdot \mathbf{c})v_i=\boxed{a_j{b}_j{u}_i+c_k{c}_k{v}_i}$

Useful Patterns

Using this conversion, we can pick up on a few patterns with vectors and matrices.

• For vectors, $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i$, as shown earlier: $\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3={\sum }_{i=1}^3{a}_i{b}_i\to \boxed{a_i{b}_i}$, by the definition of a dot product.
• For matrices, $\mathbf{AB}=A_{ik}+B_{kj}$ and $(A_{ij}{)}^T=A_{ji}$, both of which can be directly proven using vector notation and sums, but are also observable by the definitions of matrix multiplication and the transpose of a matrix.

In-Lecture Example (commutative trace of a matrix)

Be definition, $\text{Tr}(\mathbf{C})=C_{jj}$ (only the elements on the leading diagonal), and hence we can simplify $\text{Tr}(\mathbf{AB})=\text{Tr}(A_{ik}{B}_{kj})=A_{jk}{B}_{kj}$, or equivalently $\text{Tr}(\mathbf{BA})$ (by similarly evaluating and then reordering and relabelling such that the indices align).

Weekly Problem 1.4 (transpose of a matrix)

Let $A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right),B=\left(\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right)$, such that we can have $(\mathbf{AB}{)}^T={\mathbf{B}}^T{\mathbf{A}}^T$.

If we first take the left-hand side, we have $(AB{)}^T=((AB{)}_{ji})$, so $(AB{)}_{ij}^T=(AB{)}_{ji}=A_{jk}{B}_{ki}$.

On the right-hand side, we have $(B^T{A}^T{)}_{ij}=(B^T{)}_{ik}(A^T{)}_{kj}=B_{ki}{A}_{jk}=A_{jk}{B}_{ki}$.

Therefore, $(AB{)}_{ij}^T=A_{jk}{B}_{ki}=(B^T{A}^T{)}_{ij}$, so $(AB{)}^T=B^T{A}^T$. ​

Similarly, we’ll find a few tensors later that will be useful whilst evaluating expressions: the Kronecker Delta (substitution tensor) and the Alternating Tensor. First, the former…

Kronecker Delta

The Kronecker Delta, or substitution tensor, is very similar to the Dirac delta-function, just over a discrete domain rather than a continuous domain. It has the definition that ${\delta }_{ij}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$, where $i$ and $j$ can be 1, 2, or 3, which can be thought of as the identity matrix:

$${\delta }_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

Effectively, it can pick out specific indices and replace them - it replaces the repeated index with the free index. This important property can be mathematically represented as the following…

$${\delta }_{ij}{a}_j=a_i\text{or}{\delta }_{ji}{a}_i=a_j$$

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Pre-Lecture Notes from University Notes

• Tensors generalise vectors, e.g.,
  • Rank 0 tensor: scalar;
  • Rank 1 tensor: vector;
  • Rank 2 tensor: matrix;
  • All independent of the coordinate system.
• Suffix (or index) notation is used to make notation easier.
  • Like when writing vectors with unit vectors: $\mathbf{v}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$, given that each vector is real and three-dimensional.
  • For example, dot product between two vectors: $\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3={\sum }_{j=1}^3{a}_j{b}_j$.
  • A more complex example, multiplying two of these dot products: $(\mathbf{a}\cdot \mathbf{b})(\mathbf{c}\cdot \mathbf{d})=a_j{b}_j{c}_k{d}_k$.
• Rules for suffix notation:
  • Choice of index doesn’t matter, they’re just dummy indices.
  • Order of indices doesn’t matter, as long as each index appears at most twice in each term.
    • Dummy indices are such that they appear exactly twice; representing sums (like dot products).
    • Free indices are such that only appear once; representing entries of a vector (the $i$th component).
      • The number of free indices is equal to the rank tensor; i.e., a scalar has no free indices (since it creates just a dot product, the result of which is always a scalar).
• Converting to suffix notation:
  • For example, for a vector equation, $\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}=|\mathbf{a}{|}^2(\mathbf{b}\cdot \mathbf{v})\mathbf{a}$…
    1. Introduce free index, like $i$: $(\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}{)}_i=((\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})\mathbf{a}{)}_i:|\mathbf{a}|:=\sqrt{\mathbf{a}\cdot \mathbf{a}}$.
    2. Distribute free index, for each vector.: $u_i+(\mathbf{a}\cdot \mathbf{b})v_i=(\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})a_i$.
    3. Introduce dummy indices, for each scalar: $\boxed{u_i+a_j{b}_j{v}_i=a_k{a}_k{b}_{\ell }{v}_{\ell }{a}_i}$.
  • Or just an expression: $(\mathbf{a}\cdot \mathbf{b})\mathbf{u}+|\mathbf{c}{|}^2\mathbf{v}=((\mathbf{a}\cdot \mathbf{b})\mathbf{u}{)}_i+((\mathbf{c}\cdot \mathbf{c})\mathbf{v}{)}_i=(\mathbf{a}\cdot \mathbf{b})u_i+(\mathbf{c}\cdot \mathbf{c})v_i=\boxed{a_j{b}_j{u}_i+c_k{c}_k{v}_i}$
  • Similarly, you can write matrices in this notation: given $A$ and $B$ and $n\times n$ matrices, the entries of $C=AB$ can be written as $C_{ij}=A_{ik}{B}_{kj}$ - just by multiplying together and simplifying as before.
    • This can be very useful, for example to prove that (for the matrices before) $\text{Trace}(\mathbf{AB})=\text{Trace}(\mathbf{BA})$…
      • $\text{Tr}(\mathbf{C})=C_{11}+C_{22}+\cdots +C_{NN}=C_{jj}$, hence,
      • $\text{Tr}(\mathbf{AB})=\text{Tr}(A_{ik}{B}_{kj})=A_{jk}{B}_{kj}$ and $\text{Tr}(\mathbf{BA})=\text{Tr}(B_{kj{A}_{ik}})=B_{jk}{A}_{kj}$, where the indices and be re-ordered and re-labelled as discussed before, to then give $\text{Tr}(\mathbf{AB})$.
• The Kronecker delta, in this vector notation, is defined by ${\delta }_{ij}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$, where $i$ and $j$ can be 1, 2, or 3.
  • Hence, it’s really just like the identity matrix: ${\delta }_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.
  • Similar to the Dirac delta-function, although on discrete domains instead of continuous, this gives it the property that ${\delta }_{ij}{a}_j=a_i$ - absorbing the repeated index.
  • Sometimes called the substitution tensor, because it replaces the repeated index with the free index.
  • Naturally, this is symmetrical for ${\delta }_{ji}{a}_i=a_j$, due to the re-ordering/re-labelling rules, and can be proved trivially.
  • Further touched on in the next lecture.