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Diffstat (limited to 'mindmap/del operator.org')
| -rw-r--r-- | mindmap/del operator.org | 13 |
1 files changed, 11 insertions, 2 deletions
diff --git a/mindmap/del operator.org b/mindmap/del operator.org index 7410e30..3d9e2f5 100644 --- a/mindmap/del operator.org +++ b/mindmap/del operator.org @@ -86,6 +86,15 @@ vector field "perpendicular" to the divergence of the field. In fact, if you hav you can represent this field as an addition of a curl-less field and a divergence-less field. Another way to think of it is that you are measuring the strength of rotational component of the vector field about a certain axis. +** directional derivative +:PROPERTIES: +:ID: e255eb0a-246b-4a4b-8db8-ac0d15d9cc3c +:END: +The directional derivative is defined as follows: +\begin{align*} +\vec{f} \cdot \vec{\nabla} = \sum_{i=0}^{n}f_{i}\frac{\partial}{\partial x_{i}} +\end{align*} +Which represents a superposition of states which corresponds to the direction you want to take the derivative in. ** Laplacian :PROPERTIES: :ID: 65004429-a6b7-41f2-8489-07605841da3d @@ -98,6 +107,6 @@ It returns a scalar field and is the multivariable analogue to the second deriva and gradient have been described, I feel it is trivial to understand the Laplacian. ** Product Rules -The product rules pertaining to the del operator are consistent with that of linear algebra. +The product rules pertaining to the del operator are consistent with that of linear algebra and single variable derivative rules. For example, \( \vec{\nabla} \times \vec{\nabla}f = 0\). You can show this yourself quite easily, so I find no need to go over it here. -When in doubt, just assume the del works the same way as any old vector, and you will usually be correct. +When in doubt, just assume the del works the same way as any old vector except you apply the [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][product rule]], and you will usually be correct. |