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
Read more about this topic: Cylindrical Coordinate System
Famous quotes containing the words line and/or volume:
“Where were you when I laid the foundation of the earth? Tell me, if you have understanding. Who determined its measurements—surely you know! Or who stretched the line upon it? On what were its bases sunk, or who laid its cornerstone when the morning stars sang together and all the heavenly beings shouted for joy?”
—Bible: Hebrew, Job 38:4 -7.
God, to Job.
“Love is both Creator’s and Saviour’s gospel to mankind; a volume bound in rose-leaves, clasped with violets, and by the beaks of humming-birds printed with peach-juice on the leaves of lilies.”
—Herman Melville (1819–1891)