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, volume and/or elements:
“I love them
for finding what
I can’t find,
and for loving me
for the line I wrote,
and for forgetting it....”
—Denise Levertov (b. 1923)
“There is a note in the front of the volume saying that no public reading ... may be given without first getting the author’s permission. It ought to be made much more difficult to do than that.”
—Robert Benchley (1889–1945)
“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)