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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold, let $E \\to M$ be a smooth vector bundle, and let $\\nabla$ be a [[connection|5ZYRRVE8BW]] on $E$.\n\n1. Let $p \\in M$, $v \\in T_pM$, and let $s$ be a smooth [[section|WWX7H5W1RP]] of $E$. The *covariant derivative* of $s$ in the direction $v$ is $\\nabla_v s = (\\nabla_X s)(p)$, where $X$ is any smooth vector field with $X_p = v$.\n2. Let $\\gamma : I \\to M$ be a smooth curve on an interval $I$. The *covariant derivative along $\\gamma$* is the unique map $D_t$ from smooth sections of $E$ along $\\gamma$ to themselves with three properties. It is $\\R$-linear. It satisfies $D_t (f V) = f' V + f D_t V$ for every smooth $f : I \\to \\R$. And $D_t V(t) = \\nabla_{\\gamma'(t)} s$ whenever $V = s \\circ \\gamma$ for a section $s$ of $E$.\n\n## Remarks\n\nClause 1 is well defined because $\\nabla_X s$ is $C^\\infty(M)$-linear in $X$, so its value at $p$ depends only on $X_p$.\nThe existence and uniqueness in clause 2 follow by writing $V(t) = \\sum_i V^i(t) e_i(\\gamma(t))$ in a local frame: then $D_t V = \\sum_i \\big(\\dot V^i + \\sum_j \\omega^i_j(\\gamma'(t)) V^j\\big) e_i$, where $\\nabla e_j = \\sum_i \\omega^i_j \\otimes e_i$.\nExample: for the flat connection on $\\R^n$ and $E = T\\R^n$, $D_t V = \\dot V$ is the ordinary derivative of the components; so $D_t \\gamma' = \\ddot \\gamma$ is the acceleration.\nNon-example: on the round sphere $S^2 \\subset \\R^3$ with its Levi-Civita connection, $D_t \\gamma'$ is the tangential part of $\\ddot \\gamma$, not $\\ddot\\gamma$ itself.\nAlong a great circle $\\ddot \\gamma$ is normal to the sphere, so $D_t \\gamma' = 0$ although $\\ddot \\gamma \\ne 0$.\nA connection on $TM$ extends uniquely to covariant derivatives of all [[tensor fields|WX302T93TW]] that obey the product rule for $\\otimes$, commute with contractions and act on functions by $\\nabla_X f = X f$.\nIn index notation $\\nabla_i Y^k = \\partial_i Y^k + \\Gamma^k_{ij} Y^j$, often written $Y^k{}_{;i}$.\n",
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