Multimediální učebnice

Slovník pojmů

Cylindrical coordinate system

It consists of a single curvilinear basis vector φ₀ and two rectilinear basis vectors r₀, z₀. Basis vectors are mutually perpendicular (see figure). In this coordinate system, an arbitrary point is described by the vector

A = A_{r} r_{0} + A_{φ} φ_{0} + A_{z} z_{0}

Coordinate system

Differential operators are of the following form:

grad ψ = \frac{\partial ψ}{\partial r} r_{0} + \frac{1}{r} \frac{\partial ψ}{\partial φ} φ_{0} + \frac{\partial ψ}{\partial z} z_{0}

div A = \frac{1}{r} \frac{\partial}{\partial r} (r A_{r}) + \frac{1}{r} \frac{\partial A_{φ}}{\partial φ} + \frac{\partial A_{z}}{\partial z}

rot A = \frac{1}{r} | \begin{matrix} r_{0} & r φ_{0} & z_{0} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial φ} & \frac{\partial}{\partial z} \\ A_{r} & r A_{φ} & A_{z} \end{matrix} |

\nabla^{2} ψ = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial ψ}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} ψ}{\partial φ^{2}} + \frac{\partial^{2} ψ}{\partial z^{2}}

\nabla^{2} A = g r a d d i v A - r o t r o t A =

= [\nabla^{2} A_{r} - \frac{A_{r}}{r^{2}} - \frac{2}{r^{2}} \frac{\partial A_{φ}}{\partial φ}] r_{0} + [\nabla^{2} A_{φ} - \frac{A_{φ}}{r^{2}} + \frac{2}{r^{2}} \frac{\partial A_{r}}{\partial φ}] φ_{0} + \nabla^{2} A_{z} z_{0}

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