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Flux d'un champ de vecteur
\[\iint_S \vec{F}(\vec{r})\cdot \dd \vec{S} = \iint_S \dd S \vec{F}(\vec{r})\cdot \vec{n}\]
Gradient
\[\nabla f = \frac{\partial }{\partial x}\vec{e_x}+\frac{\partial }{\partial y}\vec{e_y}+\frac{\partial }{\partial z}\vec{e_z}\]
Plan tangent à une surface
L'équation est, avec un point \(M_0\) appartenant à la surface : \[\nabla f(x_0,y_0,z_0)\cdot (x-x_0, y-y_0,z-z_0) \) \(= 0\]
Plan tangent à une surface de la forme \(z \) \(= h(x,y)\)
\[z=z_0+(x-x_0)\frac{\partial}{\partial x}(x_0,y_0)+(y-y_0)\frac{\partial h}{\partial y}(x_0,y_0)\]
Condition de conservation
En écrivant \(\vec{F} \) \(= P\vec{e_x}+ Q\vec{e_y}\) : Pour un champ 2D : \(\frac{\partial P}{\partial y} \) \(= \frac{\partial Q}{\partial x}\) Pour un champ 3D : \(\frac{\partial P}{\partial y} \) \(= \frac{\partial Q}{\partial Q}\), \(\frac{\partial Q}{\partial z} \) \(= \frac{\partial R}{\partial y}\), \(\frac{\partial R}{\partial x} \) \(= \frac{\partial P}{\partial z}\)
Rotationnel
\[rot(\vec{F}) = (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}), (\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}), (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\]
Formule de Green
\[\int_C \vec{F}\cdot \dd \vec{r} = \iint_S \dd x\dd y (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\]
Divergent
\[dif(\vec{F})(x,y,z) = \frac{\partial P}{\partial x}(x,y,z) + \frac{\partial Q}{\partial y}(x,y,z) + \frac{\partial R}{\partial z}(x,y,z)\]