mth3008 final exam cheat sheet

MTH3008 Tensor Analysis, Final Exam Cheat Sheet

Created by William Fayers

Covers the whole course, with emphasis on Ch 5-7 (lectures 10-18) - the material not already in the mth3008 portfolio cheat sheet. For Ch 1-4 basics (suffix notation, Kronecker Delta / Alternating Tensor identities, differential operators, local coordinate transforms), see the portfolio sheet.

0. Quick Reference

Suffix-Notation Dictionary (condensed)

Object	Suffix form
Dot product	$a_j{b}_j$
Cross product $(a\times b{)}_i$	${ϵ}_{ijk}{a}_j{b}_k$
Matrix product $(AB{)}_{ij}$	$A_{ik}{B}_{kj}$
Trace	$A_{ii}$
Kronecker identity	${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$
$\partial x_j/\partial x_i$	${\delta }_{ij}$ (Cartesian)

The “Kill Rule”

Symmetric × Antisymmetric = 0. Used everywhere: ${ϵ}_{ijk}{S}_{jk}=0$ when $S_{jk}=S_{kj}$; ${ϵ}_{ijk}{\partial }_j{\partial }_{k}f=0$ from equality of mixed partials.

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1. Tensor Transformation Laws

Cartesian (rotation Matrix $L_{ij}$)

• Coordinates: $x_i^{\prime }=L_{ij}{x}_j$, inverse $x_i=L_{ji}{x}_j^{\prime }$.
• Orthogonality: $L_{ij}{L}_{kj}={\delta }_{ik}=L_{ji}{L}_{jk}$, $L^T=L^{-1}$, $|L|=1$.
• Basis: $L_{ij}={\mathbf{e}}_i^{\prime }\cdot {\mathbf{e}}_j$, and ${\mathbf{e}}_i^{\prime }=L_{ij}{\mathbf{e}}_j$.
• Jacobian: $\frac{\partial x_i^{\prime }}{\partial x_j}=L_{ij}$ and $\frac{\partial x_i}{\partial x_j^{\prime }}=L_{ji}$.

Rank-$r$ rule (one $L$ per free index):

$$T_{i_1\dots i_r}^{\prime }=L_{i_1{m}_1}\dots L_{i_r{m}_r}{T}_{m_1\dots m_r}$$

For rank 2 this is the matrix equation $T^{\prime }=LT{L}^T$.

Generalised Coordinates

Two kinds of transformation coefficient:

$$L_{i^{\prime }}^k=\frac{\partial x^k}{\partial x^{\prime i}}(\text{covariant}),L_k^{i^{\prime }}=\frac{\partial x^{\prime i}}{\partial x^k}(\text{contravariant})$$

with $L_{i^{\prime }}^k{L}_k^{j^{\prime }}={\delta }_i^j$.

Transformation per index level: each covariant index contracts with $L_{i^{\prime }}^{\ell }$; each contravariant index contracts with $L_{\ell }^{i^{\prime }}$.

The Quotient Rule

If $T_{ij}{b}_j$ (or similar contraction) is a tensor for every vector $\mathbf{b}$, then $T_{ij}$ is a tensor. Used to prove tensor character indirectly when the direct check is hard.

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2. Bases, Dual Bases, Metric Tensor

Dual Basis Construction

Given $\{{\mathbf{e}}_i\}$ with volume $V={\mathbf{e}}_1\cdot ({\mathbf{e}}_2\times {\mathbf{e}}_3)$:

$${\mathbf{e}}^i=\frac{{\mathbf{e}}_j\times {\mathbf{e}}_k}{V}\text{(cyclic }i,j,k)$$

Satisfies ${\mathbf{e}}^i\cdot {\mathbf{e}}_k={\delta }_k^i$. For orthonormal bases, ${\mathbf{e}}^i={\mathbf{e}}_i$.

Metric Tensor

• Covariant: $g_{ik}={\mathbf{e}}_i\cdot {\mathbf{e}}_k$.
• Contravariant: $g^{ik}={\mathbf{e}}^i\cdot {\mathbf{e}}^k=[g_{ik}{]}^{-1}$.
• Mixed: $g_i^{\cdot k}={\delta }_i^k$.

From a parametrisation: ${\mathbf{e}}_i=\partial \mathbf{r}/\partial x^i$, then $g_{ij}={\mathbf{e}}_i\cdot {\mathbf{e}}_j$.

Arc length: $(ds{)}^2=g_{ik}d{x}^{i}d{x}^k=g^{ik}d{x}_{i}d{x}_k=d{x}_{i}d{x}^i$.

Orthogonal shortcut: diagonal $g$ ⇒ $g_{ii}=h_i^2$ where $h_i=|{\mathbf{e}}_i|$, and $g^{ii}=1/g_{ii}$ (no sum).

Covariant and Contravariant Components

Any vector expands two ways: $\mathbf{A}=A^i{\mathbf{e}}_i=A_i{\mathbf{e}}^i$.

• $A^i=\mathbf{A}\cdot {\mathbf{e}}^i$ (contravariant).
• $A_i=\mathbf{A}\cdot {\mathbf{e}}_i$ (covariant).
• Raise/lower: $A_i=g_{ik}{A}^k$, $A^i=g^{ik}{A}_k$.
• Inner product: $\mathbf{A}\cdot \mathbf{B}=A^i{B}_i=A_i{B}^i=g_{ik}{A}^i{B}^k$.

Rank-2 Tensor Flavours

Four flavours of components of a second-rank tensor: $A_{ik}$, $A^{ik}$, $A_i^{\cdot k}$, $A_{\cdot k}^i$. All Associated Tensors via the metric:

$$A_{ik}=g_{i\ell }{g}_{km}{A}^{\ell m},A^{ik}=g^{i\ell }{g}^{km}{A}_{\ell m},A_i^{\cdot k}=g^{k\ell }{A}_{i\ell },A_{\cdot k}^i=g^{i\ell }{A}_{\ell k}$$

Matrix form (numerical): if $[A]$ is covariant and $[G]=[g_{ik}]$, then $[A^{ik}]=G^{-1}A{G}^{-T}$ etc.

Transformation: $A_{ik}^{\prime }=L_{i^{\prime }}^{\ell }{L}_{k^{\prime }}^m{A}_{\ell m}=(LA{L}^T{)}_{ik}$ for covariant.

Symmetry and Antisymmetry

• Symmetric: $T_{ij}=T_{ji}$. Antisymmetric: $T_{ij}=-T_{ji}$.
• Frame-independent in Cartesian rotations.
• Decomposition: $T_{ik}=\frac{1}{2}(T_{ik}+T_{ki})+\frac{1}{2}(T_{ik}-T_{ki})$.
• In generalised coords, symmetry only makes sense between indices at the same level (both upper or both lower).

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3. Tensor Algebra

Operation	Rule	Rank behaviour
Addition $A+B$	$(A+B{)}_{ik}=A_{ik}+B_{ik}$	Same rank & structure required
Outer product $A\otimes B$	$(A\otimes B{)}_{ik\ell m}=A_{ik}{B}_{\ell m}$	Rank adds; structures concatenate
Contraction	Set two indices equal (upper with lower in curvilinear)	Rank drops by 2
Inner product	Outer product + contraction	Example: $A_i{B}^i$ (scalar)

Key identities (useful for contraction proofs):

• ${\delta }_{ij}{ϵ}_{ijk}=0_k$
• ${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$
• ${ϵ}_{ijk}{ϵ}_{ijm}=2{\delta }_{km}$, ${ϵ}_{ijk}{ϵ}_{ijk}=6$

Contraction preserves tensor character via orthogonality: $L_{im}{L}_{in}={\delta }_{mn}$ (Cartesian), or via the identity $L_{i^{\prime }}^m{L}_n^{i^{\prime }}={\delta }_n^m$ (general).

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4. Covariant Differentiation

In a Local Basis that varies with position, $d{\mathbf{e}}_j\ne 0$ and ordinary partial derivatives of components don’t transform as tensors. Define the covariant derivative:

$$\boxed{A_{i,k}=\frac{\partial A_i}{\partial x^k}-{\Gamma }_{ik}^j{A}_j}\boxed{A_{\cdot k}^i=\frac{\partial A^i}{\partial x^k}+{\Gamma }_{jk}^i{A}^j}$$

Sign rule: lower index ⇒ $-\Gamma$; upper index ⇒ $+\Gamma$.

Higher-rank tensors - each index contributes one $\Gamma$ correction:

$$T_{ik,\ell }=\frac{\partial T_{ik}}{\partial x^{\ell }}-{\Gamma }_{i\ell }^m{T}_{mk}-{\Gamma }_{k\ell }^m{T}_{im}$$ $$T_{\cdot \cdot ,\ell }^{ik}=\frac{\partial T^{ik}}{\partial x^{\ell }}+{\Gamma }_{m\ell }^i{T}^{mk}+{\Gamma }_{m\ell }^k{T}^{im}$$ $$T_{\cdot k,\ell }^i=\frac{\partial T_{\cdot k}^i}{\partial x^{\ell }}+{\Gamma }_{m\ell }^i{T}_{\cdot k}^m-{\Gamma }_{k\ell }^m{T}_{\cdot m}^i$$

Reduces to ordinary partials in Cartesian (all $\Gamma =0$).

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5. Christoffel Symbols

Definitions

• Second kind: ${\Gamma }_{jk}^i={\mathbf{e}}^i\cdot {\partial }_k{\mathbf{e}}_j$, equivalently ${\partial }_k{\mathbf{e}}_j={\Gamma }_{jk}^i{\mathbf{e}}_i$.
• First kind: ${\Gamma }_{ijk}={\mathbf{e}}_i\cdot {\partial }_k{\mathbf{e}}_j$.
• Relation: ${\Gamma }_{ijk}=g_{i\ell }{\Gamma }_{jk}^{\ell }$, ${\Gamma }_{jk}^i=g^{i\ell }{\Gamma }_{\ell kj}$.

Formula from the Metric

$$\boxed{{\Gamma }_{ijk}=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^k}+\frac{\partial g_{ik}}{\partial x^j}-\frac{\partial g_{jk}}{\partial x^i}\right)}$$ $$\boxed{{\Gamma }_{jk}^i=\frac{1}{2}{g}^{i\ell }\left(\frac{\partial g_{\ell j}}{\partial x^k}+\frac{\partial g_{\ell k}}{\partial x^j}-\frac{\partial g_{jk}}{\partial x^{\ell }}\right)}$$

Orthogonal-Coordinate Shortcuts (from Problem 8.1)

If $g_{ij}=0$ for $i\ne j$:

Case	${\Gamma }_{ijk}$
$i=j=k$	$\frac{1}{2}\partial g_{ii}/\partial x^i$
$i=j\ne k$	$\frac{1}{2}\partial g_{ii}/\partial x^k$
$i\ne j=k$	$-\frac{1}{2}\partial g_{jj}/\partial x^i$
all distinct	$0$

Raise with ${\Gamma }_{jk}^i={\Gamma }_{ijk}/g_{ii}$ (no sum, diagonal).

Key Properties

• Symmetric in the last two indices: ${\Gamma }_{jk}^i={\Gamma }_{kj}^i$, ${\Gamma }_{ijk}={\Gamma }_{ikj}$.
• Vanish for a fixed basis (Cartesian).
• Not a tensor - extra inhomogeneous term in the transformation law:

$${\Gamma }_{jk}^{\prime i}=L_{\ell }^{i^{\prime }}{L}_{j^{\prime }}^m{L}_{k^{\prime }}^n{\Gamma }_{mn}^{\ell }+L_m^{i^{\prime }}{L}_{k^{\prime }}^n\frac{\partial L_{j^{\prime }}^m}{\partial x^n}$$

The “extra” term is exactly what cancels the non-tensorial part of ${\partial }_k{A}_i$ to leave $A_{i,k}$ as a tensor.

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6. Ricci’s Theorem

$$\boxed{g_{ik,\ell }=0}$$

The Metric Tensor is covariantly constant. Also $g_{,\ell }^{ik}=0$.

Useful rearrangement (for reading metric derivatives off $\Gamma$):

$$\frac{\partial g_{ik}}{\partial x^{\ell }}={\Gamma }_{ik\ell }+{\Gamma }_{ki\ell }$$

Orthogonal-coordinate consequence: ${\Gamma }_{ik\ell }=-{\Gamma }_{ki\ell }$ for $i\ne k$ (first-kind antisymmetry in the first two indices). Not true for second-kind symbols.

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7. Riemann-Christoffel & Ricci Tensors

Riemann-Christoffel Tensor

Defined by the failure of second covariant derivatives to commute:

$$\boxed{{\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i=R_{ijk}^r{A}_r}$$

Coordinate formula:

$$\boxed{R_{ijk}^r=\frac{\partial {\Gamma }_{ki}^r}{\partial x^j}-\frac{\partial {\Gamma }_{ij}^r}{\partial x^k}+{\Gamma }_{ik}^p{\Gamma }_{pj}^r-{\Gamma }_{ij}^p{\Gamma }_{pk}^r}$$

Curvature Criterion

$$R_{ijk}^r=0\text{ everywhere}⟺\text{space is Euclidean (flat)}$$

Examples worth remembering:

• Unit sphere $(a,b)$ with $g_{11}=1,g_{22}={\sin }^{2}a$: $R_{212}^1={\sin }^{2}a\ne 0$.
• Surface of revolution $\mathbf{r}=(f\cos y,f\sin y,g)$ with $(f^{\prime }{)}^2+(g^{\prime }{)}^2=1$: $R_{212}^1=-f{f}^{\prime \prime }$. Flat iff $f^{\prime \prime }=0$ (cylinder or cone).

Ricci Tensor

$$R_{ij}=R_{irj}^r$$

Rank 2, symmetric, $R_{ij}=0$ in Euclidean space. Used in general relativity; in 2D it carries all curvature information.

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8. Problem-Type Playbook

If the question asks…	Use this tool
”Show $X$ is a tensor”	Tensor Transformation Rule - apply, factor $L$s, collapse via orthogonality. Or Quotient Rule.
”Show $X$ is not a tensor”	Find an extra term in the transformation that doesn’t fit any rule (e.g. $A_i+B^i$, Christoffel symbols).
”Find $g_{ij}$, $h_i$, $ds$”	${\mathbf{e}}_i=\partial \mathbf{r}/\partial x^i$, $g_{ij}={\mathbf{e}}_i\cdot {\mathbf{e}}_j$, $h_i=\sqrt{g_{ii}}$, $d{s}^2=g_{ij}d{x}^{i}d{x}^j$.
”Find the dual basis”	${\mathbf{e}}^i=({\mathbf{e}}_j\times {\mathbf{e}}_k)/V$, cyclic. Verify with ${\mathbf{e}}^i\cdot {\mathbf{e}}_k={\delta }_k^i$.
”Find $A_i,A^i$ or $P_{ik},P^{ik},P_i^{\cdot k}$“	Dot with basis/dual, or matrix routes: $P^{\prime }=LP{L}^T$ covariant; $M^{\ast }P(M^{\ast }{)}^T$ contravariant; mixtures via metric.
”Find Christoffel symbols”	§5 formulas. In orthogonal coords use the 4-case table. Then raise with $g^{ii}=1/g_{ii}$.
”Has this space zero curvature?”	Compute $R_{ijk}^r$ - usually only $R_{212}^1$ matters in 2D. Nonzero ⇒ curved.
”Simplify $ϵϵ$ or $\delta ϵ$“	Cyclically permute shared index to 3rd position, apply $ϵϵ=\delta \delta -\delta \delta$, collapse $\delta$s. Kill rule if symmetric $\times$ antisymmetric.
”Prove a covariant derivative / Christoffel identity”	Start from definitions, use Ricci’s Theorem + symmetry ${\Gamma }_{ijk}={\Gamma }_{ikj}$ + the metric formula for $\Gamma$.

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Good luck!

The big shift from portfolio to final is moving from Cartesian + rigid bases to curvilinear + local bases: the Metric Tensor replaces the Kronecker delta, Covariant Differentiation replace partial derivatives, and Christoffel Symbols are the glue. Everything else is bookkeeping.