Cylindrical Coordinate System - Line and Volume Elements

Line and Volume Elements

See multiple integral for details of volume integration in cylindrical coordinates, and Del in cylindrical and spherical coordinates for vector calculus formulae.

In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.

The line element is

The volume element is

The surface element in a surface of constant radius (a vertical cylinder) is

The surface element in a surface of constant azimuth (a vertical half-plane) is

The surface element in a surface of constant height (a horizontal plane) is

The del operator in this system is written as

and the Laplace operator is defined by

\nabla^2 f = {1 \over \rho} {\partial \over \partial \rho} \left( \rho {\partial f \over \partial \rho} \right) + {1 \over \rho^2} {\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2 }.

Read more about this topic:  Cylindrical Coordinate System

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