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:
“There’s a line between love and fascination that’s hard to see on an evening such as this.”
—Ned Washington (1901–1976)
“So it is with books, for the most part: they work no redemption on us. The bookseller might certainly know that his customers are in no respect better for the purchase and consumption of his wares. The volume is dear at a dollar, and after to reading to weariness the lettered backs, we leave the shop with a sigh, and learn, as I did without surprise of a surly bank director, that in bank parlors they estimate all stocks of this kind as rubbish.”
—Ralph Waldo Emerson (1803–1882)
“Our institutions have a potent digestion, and may in time convert and assimilate to good all elements thrown in, however originally alien.”
—Herman Melville (1819–1891)