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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.