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        "quote": "光滑的<u id=\"concept-15-parametrized-curve\" class=\"concept-term\" data-concept-identity=\"urn:uuid:4be5393c-e464-4ede-ac10-bacd25e3b1e5\" data-concept-route=\"/concept-identities#concept-4be5393c-e464-4ede-ac10-bacd25e3b1e5\">参数曲线(parametrized curve)</u>是定义在$n$维流形$M$上的可微映射$\\gamma : ( a , b )  M$，其定义域为实轴的开区间$( a , b ) \\subseteq \\mathbb { R }$，值域包含于$M$。若$t = t _ { 0 } \\mathrm { i f } \\gamma ( t _ { 0 } ) = p$，其中$a < t _ { 0 } < b$，则称该曲线在$p$处经过。注意，参数曲线由映射构成，而非像点$\\gamma ( t ) \\in M$。将参数从$t$变为$t ^ { \\prime } = f ( t )$，其中$f : \\mathbb { R } \\to \\mathbb { R }$是单调可微函数且$a ^ { \\prime } = f ( a ) < t ^ { \\prime } < b ^ { \\prime } = f ( b )$，会将参数曲线变为$\\gamma ^ { \\prime } = \\gamma \\circ f$，但对$M$中的像点没有影响。给定$p ,$处的一个图卡$( U , \\phi ; x ^ { i } )$，开集$U$的逆像是一个开子集$\\gamma ^ { - 1 } ( U ) \\subseteq \\mathbb { R }$。设$( a _ { 1 } , b _ { 1 } )$为该集合中包含实数$t _ { 0 }$的连通分支，使得$p = \\gamma ( t _ { 0 } )$。由该图卡诱导的参数曲线$\\gamma$的“坐标表示”是光滑曲线$\\widehat { \\gamma } = \\phi \\circ \\gamma : ( a _ { 1 } , b _ { 1 } ) \\to \\mathbb { R } ^ { n }$，由$n$个实值函数$x ^ { i } = \\gamma ^ { i } ( t )$描述，其中$\\gamma ^ { i } = \\mathsf { p r } _ { i } \\circ \\phi \\circ \\gamma$。当不存在任何误解的危险时，我们通常将其简写为$x ^ { i } = x ^ { i } ( t )$（见图15.3）。在另一个图卡$( U ^ { \\prime } ; x ^ { \\prime j } )$中，表示该曲线的$n$个函数变为$x ^ { \\prime j } = \\gamma ^ { \\prime j } ( t ) = x ^ { \\prime j } \\big ( \\gamma ( t ) \\big )$。假设图卡相容，这些表示曲线的新函数仍然是光滑的，尽管参数范围$( a ^ { \\prime } , b ^ { \\prime } )$可能发生改变。",
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