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  "body": "\n## Definition\n\nLet $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]] of dimensions $m$ and $n$, and let $u \\colon M \\to N$ be a smooth map.\nRegard $du$ as a section of the bundle $T^*M \\otimes u^* TN$ over $M$, with the connection $\\nabla$ induced by the [[Levi-Civita connections|CAZYGE1T4E]] of $g$ and $h$.\nThe *tension field* of $u$ is the section of $u^* TN$\n$$\\tau(u) = \\tr_g \\nabla du = \\sum_{i=1}^m \\big( \\nabla_{e_i} du \\big)(e_i),$$\nwhere $(e_i)$ is a local orthonormal frame of $TM$.\nIn local coordinates $x^i$ on $M$ and $y^\\gamma$ on $N$, its components are\n$$\\tau(u)^\\gamma = \\Delta_g u^\\gamma + \\sum_{i,j,\\alpha,\\beta} g^{ij} \\, \\Gamma^\\gamma_{\\alpha\\beta}(u) \\, \\frac{\\partial u^\\alpha}{\\partial x^i} \\frac{\\partial u^\\beta}{\\partial x^j}.$$\nHere $\\Delta_g$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $g$, with the sign for which $\\Delta_g = \\frac{d^2}{dx^2}$ on $\\R$.\nThe $\\Gamma^\\gamma_{\\alpha\\beta}$ are the [[Christoffel symbols|FZBA1HA9ZM]] of $h$.\n\n## Remarks\n\nIf $M$ is compact without boundary and $u_s$ is a smooth family of maps with $u_0 = u$ and variation field $V = \\partial_s u_s |_{s=0}$, then the [[energy|E6HDE1DFXD]] satisfies\n$$\\frac{d}{ds}\\Big|_{s=0} E(u_s) = -\\int_M h\\big(\\tau(u), V\\big) \\, dV_g.$$\nSo $-\\tau(u)$ is the gradient of $E$, and $u$ is a [[harmonic map|KZJYG9DS3M]] exactly when $\\tau(u) = 0$.\n\nExample: let $M = \\R$ and $\\gamma \\colon \\R \\to N$ a curve; then $\\tau(\\gamma) = \\nabla_{\\dot\\gamma} \\dot\\gamma$, the covariant acceleration, so $\\tau(\\gamma) = 0$ is the geodesic equation.\nFor $\\gamma(t) = (\\cos t, \\sin t, 0)$ into the unit sphere $S^2$, a great circle, $\\tau(\\gamma) = 0$.\n\nNon-example: the curve $\\gamma(t) = (\\cos t, \\sin t)$ into $\\R^2$ with the Euclidean metric has $\\tau(\\gamma) = \\ddot\\gamma = (-\\cos t, -\\sin t) \\ne 0$, so it is not harmonic.\nThe same parametrized circle, seen as a map into the unit circle $S^1$, has tension $0$, so the tension field depends on the target.\nFor $N = \\R^n$ with the Euclidean metric the Christoffel symbols vanish and $\\tau(u) = (\\Delta_g u^1, \\dots, \\Delta_g u^n)$.\n",
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