# Basic Differential Operators

• Scalar functions: $$f\left( x \right),$$ $$f\left( {x,y,z} \right),$$ $$f\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$$
Vector function: $$\mathbf{V}\left( {P,Q,R} \right)$$
Unit vectors along the coordinate axes: $$\mathbf{i},$$ $$\mathbf{j},$$ $$\mathbf{k}$$
Operator of differentiation: $$D$$
Linear differential operator: $$L\left( D \right)$$
Direction vector: $$\ell$$
Gradient of a scalar field: $$\text {grad } f$$
Divergence of a vector field: $$\text {div } \mathbf{V}$$
Curl of a vector field: $$\text {rot } \mathbf{V}$$
Laplace operator: $$\Delta$$
D’Alembert operator: $$\require{AMSsymbols.js}\Box$$
Speed of light: $$c$$
Time: $$t$$
1. A differential operator can be considered as a generalization of the operation of differentiation. The simplest differential operator $$D$$ just means taking the first order derivative:
$$Dy\left( x \right) = {\large\frac{{dy\left( x \right)}}{{dx}}\normalsize} =$$ $$y’\left( x \right).$$
The operation $$D$$ applied $$n$$ times leads to the $$n$$th order derivative of $$y:$$
$${D^n}y\left( x \right) = {\large\frac{{{d^n}y\left( x \right)}}{{d{x^n}}}\normalsize} =$$ $${y^{\left( n \right)}}\left( x \right).$$
2. A linear differential operator is written as
$$L\left( D \right) = {D^n} + {a_1}\left( x \right){D^{n – 1}} \;+$$ $${a_2}\left( x \right){D^{n – 2}} + \ldots$$ $$+\;{a_{n – 1}}\left( x \right)D + {a_n}\left( x \right),$$
where the coefficients $${a_i}\left( x \right)$$ are functions of the variable $$x$$.
3. Theta operator
In the case of a function of one variable $$y = f\left( x \right),$$ the theta operator is given by
$$\theta = x\large\frac{d}{{dx}}\normalsize.$$
For a function of several variables $$y = f\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$$, the theta operator is written in the form
$$\theta = {x_1}{\large\frac{\partial }{{\partial {x_1}}}\normalsize} + {x_2}{\large\frac{\partial }{{\partial {x_2}}}\normalsize} + \ldots$$ $$+\; {x_n}{\large\frac{\partial }{{\partial {x_n}}}\normalsize} =$$ $$\sum\limits_{i = 1}^n {{x_i}{\large\frac{\partial }{{\partial {x_i}}}\normalsize}} .$$
4. Operator nabla
The differential operator nabla often appears in vector analysis. In the space of three variables it is defined as
$$\nabla = {\large\frac{\partial }{{\partial x}}\normalsize} \mathbf{i} + {\large\frac{\partial }{{\partial y}}\normalsize} \mathbf{j} + {\large\frac{\partial }{{\partial z}}\normalsize} \mathbf{k},$$
where $$\mathbf{i}$$, $$\mathbf{j}$$, $$\mathbf{k}$$ are the unit vectors, respectively, along the $$x$$, $$y$$ and $$z$$ axes. As a result of acting of the operator nabla on a scalar field $$f$$, we obtain the gradient of the field $$f:$$
$$\nabla f = {\large\frac{\partial f}{{\partial x}}\normalsize} \mathbf{i} + {\large\frac{\partial f}{{\partial y}}\normalsize} \mathbf{j} + {\large\frac{\partial f}{{\partial z}}\normalsize} \mathbf{k}.$$
5. The directional derivative of a scalar function can be calculated from the components of the gradient vector:
$${\large\frac{\partial f}{{\partial \ell}}\normalsize} = {\large\frac{\partial f}{{\partial x}}\normalsize} \cos\alpha \;+$$ $${\large\frac{\partial f}{{\partial y}}\normalsize} \cos\beta \;+$$ $${\large\frac{\partial f}{{\partial z}}\normalsize} \cos\gamma,$$
where the direction is determined by the unit vector $$\mathbf{\ell}\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right)$$:
$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma$$ $$= 1.$$
6. The scalar product of the operator nabla $$\nabla$$ and a vector function $$\mathbf{V}$$ is known as the divergence of the vector field $$\mathbf{V}$$:
$$\text {div }\mathbf{V} = \nabla \cdot \mathbf{V} =$$ $${\large\frac{{\partial P}}{{\partial x}}\normalsize} + {\large\frac{{\partial Q}}{{\partial y}}\normalsize} + {\large\frac{{\partial R}}{{\partial z}}\normalsize},$$ $$\mathbf{V} = \mathbf{V}\left( {P,Q,R} \right).$$
7. The vector product of the operator nabla $$\nabla$$ and a vector function $$\mathbf{V}$$ is known as the curl of the vector field $$\mathbf{V}$$:
$$\text {rot }\mathbf{V} = \nabla \times \mathbf{V} =$$ $$\left| {\begin{array}{*{20}{c}} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ {\large\frac{\partial }{{\partial x}}}\normalsize & {\large\frac{\partial }{{\partial y}}}\normalsize & {\large\frac{\partial }{{\partial z}}}\normalsize \\ P & Q & R \end{array}} \right| =$$ $$\left( {{\large\frac{{\partial R}}{{\partial y}}\normalsize} – {\large\frac{{\partial Q}}{{\partial z}}\normalsize}} \right)\mathbf{i} \;+$$ $$\left( {{\large\frac{{\partial P}}{{\partial z}}\normalsize} – {\large\frac{{\partial R}}{{\partial x}}\normalsize}} \right)\mathbf{j} \;+$$ $$\left( {{\large\frac{{\partial Q}}{{\partial x}}\normalsize} – {\large\frac{{\partial P}}{{\partial y}}\normalsize}} \right)\mathbf{k},$$ where $$\mathbf{V} = \mathbf{V}\left( {P,Q,R} \right)$$.
8. Laplace operator
The scalar product of two operators nabla forms a new scalar differential operator known as the Laplace operator or laplacian. It is denoted by the symbol $$\Delta$$:
$$\Delta = {\nabla ^2} =$$ $${\large\frac{{{\partial ^2}}}{{\partial {x^2}}}\normalsize} + {\large\frac{{{\partial ^2}}}{{\partial {y^2}}}\normalsize} + {\large\frac{{{\partial ^2}}}{{\partial {z^2}}}\normalsize}.$$
9. $$\text {div}\left( \text {rot }\mathbf{V} \right)$$ $$=\nabla \cdot \left( {\nabla \times \mathbf{V}} \right)$$ $$\equiv 0$$
10. $$\text {rot}\left( \text {grad } f \right)$$ $$=\nabla \times \left( {\nabla f} \right)$$ $$\equiv 0$$
11. $$\text {div}\left( \text {grad } f \right)$$ $$= \nabla \cdot \left( {\nabla f} \right)$$ $$\equiv \nabla^2 f$$
12. $$\text {rot}\left( \text {rot } \mathbf{V} \right)$$ $$=\text {grad}\left( \text {div } \mathbf{V} \right) – \nabla^2\mathbf{V}$$ $$=\nabla \left( {\nabla \cdot \mathbf{V}} \right) – \nabla^2\mathbf{V}$$
13. D’Alembert operator
This operator is denoted by a square $$\Box$$ and used in special relativity and other fields of physics. In the four-dimensional spacetime, it is written as
$$\Box = \large\frac{1}{{{c^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}\normalsize – \Delta ,$$
where the variable $$t$$ means time, $$c$$ is the speed of light, and $$\Delta$$ is the Laplace operator.