什麼是 conservative 的 vector field?

先講結論 gradient field $\Longleftrightarrow$ conservative

https://www.youtube.com/watch?v=76nzOtupeRc

Conservative 定義為:

這個 line integral 只跟 end points 有關, 跟怎麼走的路徑無關

A continuous field $\vec{F}$ is conservative $\Longleftrightarrow$ $\vec{F}=\nabla f$. For some differentiable scalar field $f$.

意思就是 gradient field $\Longleftrightarrow$ conservative

• Example: 證明 gravitative vector field 是 conservative

怎麼測試一個 vector field 是 conservative?

https://www.youtube.com/watch?v=ZGUvyGeNT44

首先我們把 vector field $\vec{F}$ 寫成如下的表達:

$$ \vec{F}=M(x,y,z)\vec{i}+N(x,y,z)\vec{j}+P(x,y,z)\vec{k} $$

假設 $\vec{F}$ 是 conservative 則存在一 scalar field $f$ 滿足 $\vec{F}=\nabla f$.

所以 $M=\partial f/\partial x$, $N=\partial f/\partial y$, and $P=\partial f/\partial z$.

觀察：

$$ \frac{\partial M}{\partial y}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}=\frac{\partial}{\partial x}\frac{\partial f}{\partial y}=\frac{\partial N}{\partial x} $$