Gradient : $${\bf grad}\,u = \begin{pmatrix} \displaystyle\frac{\partial\,u}{\partial x}\\[1em] \displaystyle\frac{\partial\,u}{\partial y}\\[1em] \displaystyle\frac{\partial\,u}{\partial z}\end{pmatrix}$$
Divergence : $$\text{div}\,{\bf u} = \frac{\partial\,u_x}{\partial x}+\frac{\partial\,u_y}{\partial y}+\frac{\partial\,u_z}{\partial z}$$
Rotationnel : $${\bf rot\,u} = \begin{pmatrix}\displaystyle\frac{\partial\,u_z}{\partial y}-\frac{\partial\,u_y}{\partial z}\\[1em] \displaystyle \frac{\partial\,u_x}{\partial z}-\frac{\partial\,u_z}{\partial x}\\[1em] \displaystyle \frac{\partial\,u_y}{\partial x}-\frac{\partial\,u_x}{\partial y}\end{pmatrix}$$
Laplacien : $$\Delta\,u = \frac{\partial^2\,u}{\partial x^2}+\frac{\partial^2\,u}{\partial y^2}+\frac{\partial^2\,u}{\partial z^2}$$
Gradient : $${\bf grad}\,u = \begin{pmatrix} \displaystyle\frac{\partial\,u}{\partial r}\\[1em] \displaystyle \frac{1}{r}\frac{\partial\,u}{\partial \theta}\\[1em] \displaystyle \frac{\partial\,u}{\partial z}\end{pmatrix}$$
Divergence : $$\text{div}\,{\bf u} = \frac{1}{r}\frac{\partial\,(r\,u_r)}{\partial r}+\frac{1}{r}\frac{\partial\,u_{\theta}}{\partial \theta}+\frac{\partial\,u_z}{\partial z}$$
Rotationnel : $${\bf rot\,u} = \begin{pmatrix}\displaystyle \frac{1}{r}\frac{\partial\,u_z}{\partial \theta}-\frac{\partial\,u_{\theta}}{\partial z}\\[1em] \displaystyle \frac{\partial\,u_r}{\partial z}-\frac{\partial\,u_z}{\partial r}\\[1em] \displaystyle \frac{1}{r}\left(\frac{\partial\,(r\,u_{\theta})}{\partial r}-\frac{\partial\,u_r}{\partial \theta}\right)\end{pmatrix}$$
Laplacien : $$\begin{aligned}\Delta\,u &= \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial\,u}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2\,u}{\partial \theta^2}+\frac{\partial^2\,u}{\partial z^2}\\[1em] &= \frac{\partial^2\,u}{\partial r^2} + \frac{1}{r} \frac{\partial\,u}{\partial r}+\frac{1}{r^2} \frac{\partial^2\,u}{\partial \theta^2}+\frac{\partial^2\,u}{\partial z^2}\end{aligned}$$
$$ J_n (x) = \sum_{p=0}^{\infty} \frac{(-1)^p}{p!\,(n+p)!}\,\left(\frac{x}{2}\right)^{2 p + n}$$
En particulier :
$$ \left\{\begin{aligned} &J_0(x) = \sum_{p=0}^{\infty} \frac{(-1)^p}{(p!)^2}\,\left(\frac{x}{2}\right)^{2 p} \\ &J_1 (x) = \frac{x}{2} \sum_{p=0}^{\infty} \frac{(-1)^p}{p!\,(p+1)!}\,\left(\frac{x}{2}\right)^{2 p} \end{aligned}\right.$$
Et une petite propriété intéressante : $J_0'(x) = -J_1(x)$
$${\bf u}\cdot({\bf v}\wedge{\bf w}) = {\bf v}\cdot({\bf w}\wedge{\bf u}) = {\bf w}\cdot({\bf u}\wedge{\bf v})$$
$${\bf u}\wedge({\bf v}\wedge{\bf w}) = ({\bf w}\wedge{\bf v})\wedge{\bf u} = {\bf v}({\bf u}\cdot{\bf w})-{\bf w}({\bf u}\cdot{\bf v})$$
$$\text{div}\,\left({\bf u}\wedge{\bf v}\right) = ({\bf rot\,u})\cdot{\bf v}-{\bf u}\cdot({\bf rot\,v})$$
$${\bf rot}({\bf rot\,u}) = {\bf grad}(\text{div}\,{\bf u}) - {\bf \Delta\,u}$$
grad-div : $$ \text{div}(u\,{\bf v}) = ({\bf grad}\,u)\cdot{\bf v} + (\text{div}\,{\bf v})\,u $$
rot-rot : $$\text{div}({\bf u}\wedge{\bf v}) = {\bf v}\cdot{\bf rot\,u} - {\bf u}\cdot{\bf rot\,v}$$
$$\text{com}\,{}^{\operatorname t}{\bf M} = {}^{\operatorname t}\text{com}\,{\bf M}$$
$${\bf M}\cdot{\bf a}\wedge{\bf M}\cdot{\bf b} = \text{com}\,{\bf M} \cdot ({\bf a}\wedge{\bf b})$$
$$\text{Soit :}~ ~{\bf M} = \begin{pmatrix} {\bf c_1} & {\bf c_2} & {\bf c_3}\end{pmatrix} , ~ ~\text{alors :}~ ~ \det\,{\bf M} = {\bf c_1}\cdot ({\bf c_2}\wedge{\bf c_3})$$