\documentclass{article} \usepackage{tensorstyles} \tensorsset{\tensor}[preset=math, preset*=einstein] \usepackage{keytheorems} % SETUP % Plain style: corollary, lemma, conjecture \newkeytheorem{corollary}[name=Corollary, style=plain] \newkeytheorem{lemma}[name=Lemma, style=plain] \newkeytheorem{conjecture}[name=Conjecture, style=plain] % Break style: theorem \newkeytheorem{theorem}[name=Theorem, style=plain] \newkeytheorem{paradox}[name=Paradox, style=plain] %\newkeytheorem{proof}[name=Proof, style=break] % Definition style: property, proposition, contraposition, consequence, hypothesis \newkeytheorem{property}[name=Property, style=definition] \newkeytheorem{proposition}[name=Proposition, style=definition] \newkeytheorem{contraposition}[name=Contraposition, style=definition] \newkeytheorem{consequence}[name=Consequence, style=definition] \newkeytheorem{hypothesis}[name=Hypothesis, style=definition] % Defbreak style: Definition, reminder \newkeytheorem{definition}[name=Definition, style=definition] \newkeytheorem{reminder}[name=Reminder, style=definition] % Remark style: remark, example, counterexample, exercise \newkeytheorem{remark}[name=Remark, style=remark] \newkeytheorem{example}[name=Example, style=remark] \newkeytheorem{counterexample}[name=CounterExample, style=remark] \begin{document} % https://grinfeld.org/books/An-Introduction-To-Tensor-Calculus/Chapter7.html % https://www.underleaf.ai/learn/latex/tensors % https://rodolphe-vaillant.fr/images/tmp/cours_tenseurs_ups.pdf Tensors are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions. \LaTeX{} provides excellent support for typesetting tensor notation, which is essential in physics, engineering, and advanced mathematics. This guide covers the essential \LaTeX{} commands for working with tensors. \section{Basic notion} \begin{definition} Let $V$ be a $\mathbb{K}$-vectorial space. A form (or linear map) on $V$ is an application from $V$ to $\mathbb{K}$. For a vector space $V$, that we note $EV*$ it's dual space, which is the vector space of it's linear map on $V$. \end{definition} \begin{definition} A tensor of order $(p,q)$, noted $T$, is a multi-linear map defined as \begin{equation} T: \underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ copies}}}\times \underbrace {V\times \dots \times V} _{q{\text{ copies}}}\rightarrow \mathbb{K} \end{equation} \end{definition} \begin{remark} By applying a multi-linear map $T$ of type $(p, q)$ to a basis ${e_j}$ for $V$ and a canonical co-basis ${\varepsilon_i}$ for $V^*$, \begin{equation} \tensor[preset=einstein, collapsed-indices=true]{xX}{T}[_{j_{1}\ldots j_{q}}^{i_{1}\ldots i_{p}}] \equiv T\left({\boldsymbol {\varepsilon }}^{i_{1}},\ldots ,{\boldsymbol {\varepsilon }}^{i_{p}},\mathbf{e}_{j_{1}},\ldots ,\mathbf{e}_{j_{q}}\right), \end{equation} a $(p + q)$-dimensional array of components can be obtained. \end{remark} % Basic Tensor Notation % Tensors are typically represented using indices to denote their components: % Tensor Symbols % T, \mathbf{T}, \mathsf{T}, \mathcal{T}, \mathbb{T} % T,T,T,T,T % T,T,T,T,T % Different ways to represent tensor symbols in LaTeX. % Tensor Components with Indices % T^{i}_{j}, \quad T^{ij}_{k}, \quad T^{i_1 i_2 \ldots i_n}_{j_1 j_2 \ldots j_m} % Tji,Tkij,Tj1j2…jmi1i2…in % Tji​,Tkij​,Tj1​j2​…jm​i1​i2​…in​​ % Tensor components with superscript (contravariant) and subscript (covariant) indices. % Tensor Rank % \text{Rank } (m,n) \text{ tensor: } T^{i_1 i_2 \ldots i_m}_{j_1 j_2 \ldots j_n} % Rank (m,n) tensor: Tj1j2…jni1i2…im % Rank (m,n) tensor: Tj1​j2​…jn​i1​i2​…im​​ % A tensor of rank (m,n) has m contravariant indices and n covariant indices. % Einstein Summation Convention % The Einstein summation convention is commonly used with tensors, where repeated indices imply summation: % Implicit Summation % A^i B_i = \sum_{i=1}^n A^i B_i % AiBi=∑i=1nAiBi % AiBi​=i=1∑n​AiBi​ % When an index appears once as a superscript and once as a subscript, summation is implied. % Matrix Multiplication as Tensor Contraction % C^i_j = A^i_k B^k_j = \sum_{k=1}^n A^i_k B^k_j % Cji=AkiBjk=∑k=1nAkiBjk % Cji​=Aki​Bjk​=k=1∑n​Aki​Bjk​ % Matrix multiplication expressed as tensor contraction using Einstein notation. % Free Indices % D^{ij}_k = A^i_l B^j_m C^{lm}_k % Dkij=AliBmjCklm % Dkij​=Ali​Bmj​Cklm​ % Indices that appear only once (i, j, k) are free indices and represent components of the resulting tensor. % Common Tensors in Physics % Metric Tensor % g_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} % gμν=(−1000010000100001) % gμν​= % ​−1000​0100​0010​0001​ % ​ % The Minkowski metric tensor used in special relativity. % Kronecker Delta % \delta^i_j = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} % δji={1if i=j0if i≠j % δji​={10​if i=jif i=j​ % The Kronecker delta is a rank (1,1) tensor that acts as an identity operator. % Levi-Civita Symbol % \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{if any index is repeated} \end{cases} % εijk={+1if (i,j,k) is an even permutation of (1,2,3)−1if (i,j,k) is an odd permutation of (1,2,3)0if any index is repeated % εijk​=⎩ % ⎨ % ⎧​+1−10​if (i,j,k) is an even permutation of (1,2,3)if (i,j,k) is an odd permutation of (1,2,3)if any index is repeated​ % The Levi-Civita symbol is used for cross products and determinants. % Tensor Operations % Tensor Addition % C^{ij}_k = A^{ij}_k + B^{ij}_k % Ckij=Akij+Bkij % Ckij​=Akij​+Bkij​ % Tensors of the same rank can be added component-wise. % Tensor Contraction % A^i_i = \sum_{i=1}^n A^i_i % Aii=∑i=1nAii % Aii​=i=1∑n​Aii​ % Contraction of a tensor by setting a contravariant and covariant index equal. % Tensor Product % C^{ij}_{kl} = A^i_k \otimes B^j_l % Cklij=Aki⊗Blj % Cklij​=Aki​⊗Blj​ % The tensor product combines two tensors into a higher-rank tensor. % Raising and Lowering Indices % The metric tensor can be used to raise or lower indices: % Raising an Index % A^{\mu} = g^{\mu\nu}A_{\nu} % Aμ=gμνAν % Aμ=gμνAν​ % Using the inverse metric tensor to raise an index. % Lowering an Index % A_{\mu} = g_{\mu\nu}A^{\nu} % Aμ=gμνAν % Aμ​=gμν​Aν % Using the metric tensor to lower an index. % Covariant Derivatives % Christoffel Symbols % \Gamma^{\lambda}_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}\left(\frac{\partial g_{\rho\mu}}{\partial x^{\nu}} + \frac{\partial g_{\rho\nu}}{\partial x^{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\right) % Γμνλ=12gλρ(∂gρμ∂xν+∂gρν∂xμ−∂gμν∂xρ) % Γμνλ​=21​gλρ(∂xν∂gρμ​​+∂xμ∂gρν​​−∂xρ∂gμν​​) % Covariant Derivative of a Vector % \nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu\lambda}V^{\lambda} % ∇μVν=∂μVν+ΓμλνVλ % ∇μ​Vν=∂μ​Vν+Γμλν​Vλ % The covariant derivative generalizes the partial derivative to curved spaces. % Covariant Derivative of a Covector % \nabla_{\mu}V_{\nu} = \partial_{\mu}V_{\nu} - \Gamma^{\lambda}_{\mu\nu}V_{\lambda} % ∇μVν=∂μVν−ΓμνλVλ % ∇μ​Vν​=∂μ​Vν​−Γμνλ​Vλ​ % Advanced Tensor Notation % R^{\rho}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}_{\mu\sigma} + \Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma} % Rσμνρ=∂μΓνσρ−∂νΓμσρ+ΓμλρΓνσλ−ΓνλρΓμσλ % Rσμνρ​=∂μ​Γνσρ​−∂ν​Γμσρ​+Γμλρ​Γνσλ​−Γνλρ​Γμσλ​ % The Riemann curvature tensor. % R_{\mu\nu} = R^{\lambda}_{\mu\lambda\nu} % Rμν=Rμλνλ % Rμν​=Rμλνλ​ % The Ricci tensor, a contraction of the Riemann tensor. % R = g^{\mu\nu}R_{\mu\nu} % R=gμνRμν % R=gμνRμν​ % The Ricci scalar, a contraction of the Ricci tensor. % G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R % Gμν=Rμν−12gμνR % Gμν​=Rμν​−21​gμν​R % The Einstein tensor, used in general relativity. \end{document}