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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, let $T_pM$ be the tangent space at $p \\in M$, and let $T_p^*M$ be its dual space.\nLet $k, l \\ge 0$ be integers.\nA *tensor of type $(k, l)$ at $p$* is an element of the [[tensor product|0BF2DE2SYK]]\n$$T^k_l(T_pM) = \\underbrace{T_pM \\otimes \\cdots \\otimes T_pM}_{k} \\otimes \\underbrace{T_p^*M \\otimes \\cdots \\otimes T_p^*M}_{l}.$$\nThese spaces form a smooth vector bundle $T^k_l M \\to M$, built from the [[tangent bundle|WY6K3P4YCG]] as $TM$ is.\nA *tensor field of type $(k, l)$* on $M$ is a smooth [[section|WWX7H5W1RP]] of $T^k_l M$.\nIn a chart with coordinates $x^1, \\dots, x^n$, a tensor field $T$ is written as follows, with summation over repeated indices.\n$$T = T^{i_1 \\cdots i_k}_{j_1 \\cdots j_l} \\, \\partial_{i_1} \\otimes \\cdots \\otimes \\partial_{i_k} \\otimes dx^{j_1} \\otimes \\cdots \\otimes dx^{j_l}.$$\nSmoothness means that the component functions $T^{i_1 \\cdots i_k}_{j_1 \\cdots j_l}$ are smooth in every chart.\n\n## Remarks\n\nExample: on $\\R^2$, $g = dx \\otimes dx + dy \\otimes dy$ is a tensor field of type $(0, 2)$; its components are the constants $g_{11} = g_{22} = 1$ and $g_{12} = g_{21} = 0$, which are smooth.\nA tensor field of type $(1, 0)$ is a vector field, and a tensor field of type $(0, 0)$ is a smooth function.\nNon-example: the Christoffel symbols of a connection are not the components of a tensor field.\nUnder a change of chart they pick up a term with second derivatives of the coordinate change.\nA tensor field of type $(0, l)$ is the same as a map $(X_1, \\dots, X_l) \\mapsto T(X_1, \\dots, X_l)$ from $l$ vector fields to smooth functions that is $C^\\infty(M)$-linear in each argument.\nThis criterion is the standard way to check that an expression is tensorial.\n",
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