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Cartesian coordinate system

It consists of three rectilinear mutually perpendicular basis vectors x₀, y₀, z₀ (see figure). In this system, an arbitrary point is described by the vector

A = A_{x} x_{0} + A_{y} y_{0} + A_{z} z_{0}

Coordinate system

Differential operators are described by the following relations:

grad ψ = \nabla ψ = \frac{\partial ψ}{\partial x} x_{0} + \frac{\partial ψ}{\partial y} y_{0} + \frac{\partial ψ}{\partial z} z_{0}

div A = \nabla \cdot A = \frac{\partial A_{x}}{\partial x} + \frac{\partial A_{y}}{\partial y} + \frac{\partial A_{z}}{\partial z}

rot A = \nabla \times A = | \begin{matrix} x_{0} & y_{0} & z_{0} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{matrix} |

\nabla^{2} ψ = \nabla \cdot \nabla ψ = \frac{\partial^{2} ψ}{\partial x^{2}} + \frac{\partial^{2} ψ}{\partial y^{2}} + \frac{\partial^{2} ψ}{\partial z^{2}}

\nabla^{2} A = \nabla^{2} A_{x} x_{0} + \nabla^{2} A_{y} y_{0} + \nabla^{2} A_{z} z_{0} =

= [\frac{\partial^{2} A_{x}}{\partial x^{2}} + \frac{\partial^{2} A_{x}}{\partial y^{2}} + \frac{\partial^{2} A_{x}}{\partial z^{2}}] x_{0} + [\frac{\partial^{2} A_{y}}{\partial x^{2}} + \frac{\partial^{2} A_{y}}{\partial y^{2}} + \frac{\partial^{2} A_{y}}{\partial z^{2}}] y_{0} + [\frac{\partial^{2} A_{z}}{\partial x^{2}} + \frac{\partial^{2} A_{z}}{\partial y^{2}} + \frac{\partial^{2} A_{z}}{\partial z^{2}}] z_{0}

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