Derivative, Gradient, Jacobian, Hessian, Laplacian
Just some basic notations
Derivative
$$ f^{\prime} (x) = \frac {df(x)} {dx} $$
Gradient
Generalize the derivative to the multivariate functions.
The first order derivative of a multivariate functions.
$$ \nabla f=\left[\frac{\partial f\left(x_{1}, x_{2}, x_{3}\right)}{\partial x_{1}}, \frac{\partial f\left(x_{1}, x_{2}, x_{3}\right)}{\partial x_{2}}, \frac{\partial f\left(x_{1}, x_{2}, x_{3}\right)}{\partial x_{3}}\right] $$
Jacobian
a generalization of the derivate operator to the vector-valued functions
$$ J=\left(\begin{array}{cccc} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}} & \cdots & \frac{\partial f_{1}}{\partial x_{n}} \cr \frac{\partial f_{2}}{\partial x_{1}} & \frac{\partial f_{2}}{\partial x_{2}} & \cdots & \frac{\partial f_{2}}{\partial x_{n}} \cr \vdots & \vdots & \ddots & \vdots \cr \frac{\partial f_{m}}{\partial x_{1}} & \frac{\partial f_{m}}{\partial x_{2}} & \cdots & \frac{\partial f_{m}}{\partial x_{n}} \end{array}\right) $$
Hessian
the second order derivative of a multivariate function
$$ H=\left(\begin{array}{cccc} \frac{\partial^{2} f}{\partial x_{1}^{2}} & \frac{\partial^{2} f}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{1} \partial x_{n}} \cr \frac{\partial^{2} f}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{2} \partial x_{n}} \cr \vdots & \vdots & \ddots & \vdots \cr \frac{\partial^{2} f}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f}{\partial x_{n} \partial x_{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{n}^{2}} \end{array}\right) $$
Laplacian
The trace of the Hessian matrix is known as the Laplacian operator denoted by $\nabla^2$:
$$ \nabla^2 f = \operatorname{trace}(H) = \frac{\partial^{2} f}{\partial x_{1}^{2}}+\frac{\partial^{2} f}{\partial x_{2}^{2}}+\cdots+\frac{\partial^{2} f}{\partial x_{n}^{2}} $$
Laplace’s equation
The second order partial differential equation
In rectangular coordinates,
$$ \nabla^{2} f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0 $$
In cylindrical coordinates,
$$ \nabla^{2} f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \phi^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0 $$
In spherical coordinates, using the $(r, \theta, \varphi)$ convention,
$$ \nabla^{2} f=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \varphi^{2}}=0 $$
More generally, in curvilinear coordinates,
$$ \nabla^{2} f=\frac{\partial}{\partial \xi^{j}}\left(\frac{\partial f}{\partial \xi^{k}} g^{k j}\right)+\frac{\partial f}{\partial \xi^{j}} g^{j m} \Gamma_{m n}^{n}=0 $$
or
$$ \nabla^{2} f=\frac{1}{\sqrt{|g|}} \frac{\partial}{\partial \xi^{i}}\left(\sqrt{|g|} g^{i j} \frac{\partial f}{\partial \xi^{j}}\right)=0, \quad\left(g=\operatorname{det}\left{g_{i j}\right}\right) $$