Gradiant
Given the function \(f(x,y,z)\), the variation of \(f\) is written: \(df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz\) is equal to the scalar product of the displacement vector \(\overrightarrow{dM}(dx,dy,dz)\) and the vector coordinates \((\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\)
Hence: \(df=\overrightarrow{grad}f.\overrightarrow{dM} \text{ with }\overrightarrow{grad}f(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\)
Vector characteristics
Surface normal iso-\(f\)
Directed in the direction of increasing \(f\)
Of Cartesian coordinates \((\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\)
Contact details
Cartesians: \((x,y,z)\) Displacement vector \((dx,dy,dz) \rightarrow \overrightarrow{grad}f (\frac{\partial}{\\partial x},\frac{\ partial}{\partial y},\frac{\partial}{\partial z})\)
Cylindrical: \((r,\theta,z)\) Displacement vector \((dr,rd\theta,dz) \rightarrow \overrightarrow{grad}f (\frac{\partial}{\partial r},\frac {\partial}{r\partial\theta},\frac{\partial}{\partial z})\)
Spherical: \((r,\theta,\phi)\) Displacement vector \((dr,r.sin\phi d\theta,rd\phi) \rightarrow \overrightarrow{grad}f (\frac{\partial }{\partial r},\frac{\partial}{r.sin\phi.\partial\theta},\frac{\partial}{r\partial\phi})\)
Nabla
Partial derivative operator (not a vector)
Consider an orthonormal basis \((\overrightarrow{i},\overrightarrow{j},\overrightarrow{k})\):
$$\overrightarrow{grad}U = \overrightarrow{\nabla}U = (\frac{\partial}{\partial x}\overrightarrow{i},\frac{\partial}{\partial y}\overrightarrow{j },\frac{\partial}{\partial z}\overrightarrow{k})U = \frac{\partial U}{\partial x}\overrightarrow{i},\frac{\partial U}{\partial y }\overrightarrow{j},\frac{\partial U}{\partial z}\overrightarrow{k}$$
Divergence
Let \( \overrightarrow{E}=\overrightarrow{E}(M) = E_x \overrightarrow{i} + E_y \overrightarrow{j} +E_z \overrightarrow{k} \rightarrow \) be a vector field $$div \overrightarrow{E} = \overrightarrow{\nabla}\overrightarrow{E} = (\frac{\partial}{\partial x}\overrightarrow{i},\frac{\partial}{\partial y} \overrightarrow{j},\frac{\partial}{\partial z}\overrightarrow{k}).(E_x \overrightarrow{i} + E_y \overrightarrow{j} +E_z \overrightarrow{k}) = \frac{ \partial U}{\partial x}\overrightarrow{i},\frac{\partial U}{\partial y}\overrightarrow{j},\frac{\partial U}{\partial z}\overrightarrow{k} $$ $$div \overrightarrow{E} = \overrightarrow{\nabla}\overrightarrow{E} = \frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{ \partial E_z}{\partial z}$$
Rotational
$$\overrightarrow{rot}\overrightarrow{E} = \overrightarrow{\nabla} \wedge \overrightarrow{E} = \left| \begin{array}{cc} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array} \right| \wedge \left| \begin{array}{cc} E_x \\ E_y \\ E_z \end{array} \right| = \left| \begin{array}{cc} \frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z} \\ \frac{\partial E_x}{\partial z}-\frac {\partial E_z}{\partial x} \\ \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y} \end{array} \right|$$
Laplacian
Scalar
$$ \triangle P = \overrightarrow{\nabla}^2 P = \overrightarrow{\nabla}(\overrightarrow{\nabla} P) = \frac{\partial^2 P}{\\partial x^2}, \frac{\partial^2 P}{\partial y^2},\frac{\partial^2 P}{\partial z^2} \text{ with } P(x,y,z) \text{ one vector field}$$
Vectorial
$$\overrightarrow{\triangle} \overrightarrow{u} = \overrightarrow{\nabla}^2 \overrightarrow{u} = \left| \begin{array}{cc} \frac{\partial^2 u_x}{\partial x^2}+\frac{\partial^2 u_x}{\partial y^2}+\frac{\partial^2 u_x }{\partial z^2} \\ \frac{\partial^2 u_y}{\partial x^2}+\frac{\partial^2 u_y}{\partial y^2}+\frac{\partial^ 2 u_y}{\partial z^2} \\ \frac{\partial^2 u_z}{\partial x^2}+\frac{\partial^2 u_z}{\partial y^2}+\frac{\partial^2 u_z}{\partial z^2} \end{array} \right|$$
Relations
$$div(\overrightarrow{grad})= \text{Scalar Laplacian} $$ $$div(rot)=0$$ $$rot(rot)=\overrightarrow{grad}(div)-\text{Vectorial Laplacian}$$ $$\text{Vector Laplacian} \leftrightarrow \triangle \overrightarrow{V} = \overrightarrow{grad}(div\overrightarrow{V})-\overrightarrow{rot}(\overrightarrow{rot}\overrightarrow{V})$ $ $$rot(\overrightarrow{grad}) = 0 $$ $$\overrightarrow{a} \wedge (\overrightarrow{b} \wedge \overrightarrow{c})= \overrightarrow{b}.(\overrightarrow{a}.\overrightarrow{c})-\overrightarrow{c}. (\overrightarrow{a}.\overrightarrow{b})=(\overrightarrow{a}.\overrightarrow{c}).\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b}).\overrightarrow {c}$$