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

Famous quotes containing the words line and, line, volume and/or elements:

    “This wild night, gathering the washing as if it were flowers
    animal vines twisting over the line and
    slapping my face lightly, soundless merriment
    in the gesticulations of shirtsleeves ...”
    —Denise Levertov (b. 1923)

    “For a woman to get a rewarding sense of total creation by way of the multiple monotonous chores that are her daily lot would be as irrational as for an assembly line worker to rejoice that he had created an automobile because he tightened a bolt.”
    —Edith Mendel Stern (1901–1975)

    “I dare say I am compelled, unconsciously compelled, now to write volume after volume, as in past years I was compelled to go to sea, voyage after voyage. Leaves must follow upon each other as leagues used to follow in the days gone by, on and on to the appointed end, which, being Truth itself, is One—one for all men and for all occupations.”
    —Joseph Conrad (1857–1924)

    “The Laws of Nature are just, but terrible. There is no weak mercy in them. Cause and consequence are inseparable and inevitable. The elements have no forbearance. The fire burns, the water drowns, the air consumes, the earth buries. And perhaps it would be well for our race if the punishment of crimes against the Laws of Man were as inevitable as the punishment of crimes against the Laws of Nature—were Man as unerring in his judgments as Nature.”
    —Henry Wadsworth Longfellow (1807–1882)