diff options
Diffstat (limited to 'mindmap/conservative force.org')
| -rw-r--r-- | mindmap/conservative force.org | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/mindmap/conservative force.org b/mindmap/conservative force.org index 9b01117..ca50664 100644 --- a/mindmap/conservative force.org +++ b/mindmap/conservative force.org @@ -26,11 +26,11 @@ integrate over \(\vec{f}\), we define \(V(\vec{r})\) such that: \int_{\vec{a}}^{\vec{b}}\vec{f} \cdot d\vec{l} = V(\vec{a}) - V(\vec{b}) \end{align*} because it is path independent, we do not need to consider the infinite paths between \(\vec{a}\) and \(\vec{b}\), which -allows us to define this function \(V(\vec{r})\). Then by the fundamental theorem of calculus: +allows us to define this function \(V(\vec{r})\). Then by the fundamental theorem of calculus, using the [[id:3587c3b4-c3d8-4ff1-b0ba-8eecb1ef0e4c][Gradient]]: \begin{align*} \vec{f} = -\vec{\nabla}V \end{align*} -Therefore, conservative forces can be represented by a scalar field. Now taking the curl of both sides we get: +Therefore, conservative forces can be represented by a scalar field. Now taking the [[id:b25e0e44-c764-4f0a-a5ad-7f9d79c7660d][Curl]] of both sides we get: \begin{align*} \vec{\nabla} \times \vec{f} = 0 \end{align*} |