# 梯度定理

${\displaystyle \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{\gamma [\mathbf {p} ,\,\mathbf {q} ]}\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .}$

## 证明

${\displaystyle \varphi }$ 是个从${\displaystyle \scriptstyle \mathbb {R} ^{n}}$ 中的开集${\displaystyle U}$ ${\displaystyle \scriptstyle \mathbb {R} }$ 可微函数，设${\displaystyle r}$ 闭区间${\displaystyle [a,b]}$ ${\displaystyle U}$ 的可微函数，那么由多元复合函数求导法则复合函数${\displaystyle \varphi \circ r}$ 在闭区间${\displaystyle [a,b]}$ 上可微，并且对所有${\displaystyle t\in [a,b]}$

${\displaystyle {\frac {d}{dt}}(\varphi \circ \mathbf {r} )(t)=\nabla \varphi (\mathbf {r} (t))\cdot \mathbf {r} '(t)}$

${\displaystyle \varphi }$ 的定义域${\displaystyle U}$ 中含有从pq的可微曲线γ，定向为从pq。设${\displaystyle {\mathbf {r} }(t)}$ 是γ的参数化（其中${\displaystyle t\in [a,b]}$ ），那么上面的式子说明

{\displaystyle {\begin{aligned}\int _{\gamma }\nabla \varphi (\mathbf {u} )\cdot d\mathbf {u} &=\int _{a}^{b}\nabla \varphi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt\\&=\int _{a}^{b}{\frac {d}{dt}}\varphi (\mathbf {r} (t))dt=\varphi (\mathbf {r} (b))-\varphi (\mathbf {r} (a))=\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)\end{aligned}}}

## 参考文献

1. Williamson, Richard and Trotter, Hale. Multivariable Mathematics. Pearson Education, Inc. 2004.