# Formulas and Tables

Calculus# Basic Differential Operators

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: \(\ell\)

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: \(t\)

- 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 \(y:\)

\({D^n}y\left( x \right) = {\large\frac{{{d^n}y\left( x \right)}}{{d{x^n}}}\normalsize} =\) \({y^{\left( n \right)}}\left( x \right).\) - 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 \(x\). - 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

\(\theta = {x_1}{\large\frac{\partial }{{\partial {x_1}}}\normalsize} + {x_2}{\large\frac{\partial }{{\partial {x_2}}}\normalsize} + \ldots \) \(+\; {x_n}{\large\frac{\partial }{{\partial {x_n}}}\normalsize} =\) \( \sum\limits_{i = 1}^n {{x_i}{\large\frac{\partial }{{\partial {x_i}}}\normalsize}} .\) - 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 \(f:\)

\(\nabla f = {\large\frac{\partial f}{{\partial x}}\normalsize} \mathbf{i} + {\large\frac{\partial f}{{\partial y}}\normalsize} \mathbf{j} + {\large\frac{\partial f}{{\partial z}}\normalsize} \mathbf{k}.\) - 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 \(\mathbf{\ell}\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right)\):

\({\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma\) \( = 1.\) - The scalar product of the operator nabla \(\nabla\) and a vector function \(\mathbf{V}\) is known as the divergence of the vector field \(\mathbf{V}\):

\(\text {div }\mathbf{V} = \nabla \cdot \mathbf{V} =\) \({\large\frac{{\partial P}}{{\partial x}}\normalsize} + {\large\frac{{\partial Q}}{{\partial y}}\normalsize} + {\large\frac{{\partial R}}{{\partial z}}\normalsize},\) \(\mathbf{V} = \mathbf{V}\left( {P,Q,R} \right).\) - 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 \(\mathbf{V} = \mathbf{V}\left( {P,Q,R} \right)\). - 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 \(\Delta\):

\(\Delta = {\nabla ^2} =\) \({\large\frac{{{\partial ^2}}}{{\partial {x^2}}}\normalsize} + {\large\frac{{{\partial ^2}}}{{\partial {y^2}}}\normalsize} + {\large\frac{{{\partial ^2}}}{{\partial {z^2}}}\normalsize}.\) - \(\text {div}\left( \text {rot }\mathbf{V} \right)\) \(=\nabla \cdot \left( {\nabla \times \mathbf{V}} \right) \) \(\equiv 0\)
- \(\text {rot}\left( \text {grad } f \right) \) \(=\nabla \times \left( {\nabla f} \right) \) \(\equiv 0\)
- \(\text {div}\left( \text {grad } f \right) \) \(= \nabla \cdot \left( {\nabla f} \right) \) \(\equiv \nabla^2 f\)
- \(\text {rot}\left( \text {rot } \mathbf{V} \right) \) \(=\text {grad}\left( \text {div } \mathbf{V} \right) – \nabla^2\mathbf{V} \) \(=\nabla \left( {\nabla \cdot \mathbf{V}} \right) – \nabla^2\mathbf{V}\)
- 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.