Vector integration - Revision history

DavidB4-bot: /* Volume integrals */Category

2016-07-29T20:18:27Z

Volume integrals: Category

DavidB4-bot: /* Volume integrals */clean up & uniformity

2016-07-13T21:20:05Z

Volume integrals: clean up & uniformity

Aschlafly: added example for surface integral

2009-12-27T21:32:43Z

added example for surface integral

Aschlafly: added surface integrals

2009-12-27T20:38:15Z

added surface integrals

Aschlafly: added section on line integrals

2009-12-27T20:16:40Z

added section on line integrals

Aschlafly: a start

2009-12-27T19:59:50Z

a start

New page

'''Vector integration''' refers to four types of integrals of [[vectors]]:

*ordinary integrals, indefinite or definite
*line integrals, which sums the vector values with respect to points on a line
*surface integrals, which sums the vector values with respect to an area (performed as a double integral for each coordinate)
*volume integrals, which sums the vector values with respect to a volume (performed as a triple integral for each coordinate)

[[Category:vector analysis]]
[[Category:calculus]]
[[Category:mathematics]]

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 The volume integral is also known as the "space integral."
-[[Category:Vector analysis]]
+[[Category:Vector Analysis]]
 [[Category:Calculus]]
 [[Category:Mathematics]]

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 The volume integral is also known as the "space integral."
-[[Category:vector analysis]]
+[[Category:Vector analysis]]
-[[Category:calculus]]
+[[Category:Calculus]]
-[[Category:mathematics]]
+[[Category:Mathematics]]

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 == Line integrals ==
-The line integral of a vector is calculated by summing the [[dot product]] of the vector function with the position vector along the curve.  In physics, an example of a line integral is the work performed by a vector force along an object as it moves along the line or path.
+The line integral of a vector is the summation of the [[dot product]] of the vector function with the position vector along the curve.  In physics, an example of a line integral is the work performed by a vector force along an object as it moves along the line or path.
 If the curve ''C'' is simple and closed (like a circle), then the value of the line integral is the "circulation" of the vector function about ''C'', as in the case of a vector function that represents the velocity of a fluid.
 The line integral of a "[[conservative vector field]]" around any closed curve is 0.  The line integral of a conservative vector field from points P<sub>1</sub> to P<sub>2</sub> is independent of the curve chosen between those two points.  If a vector function can be represented as the gradient of a single-valued, continuous function (as in the case of potential energy), then the vector function must be conservative and satisfy the above two conditions.  The curl of such a vector function must then be zero.
 [[Category:vector analysis]]
 [[Category:calculus]]
 [[Category:mathematics]]

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 *ordinary integrals, indefinite or definite
-*line integrals, which sums the vector values with respect to points on a line
+*line integrals: the vector values with respect to points on a line
-*surface integrals, which sums the vector values with respect to an area (performed as a double integral for each coordinate)
+*surface integrals: sum the vector values with respect to an area (performed as a double integral for each coordinate)
-*volume integrals, which sums the vector values with respect to a volume (performed as a triple integral for each coordinate)
+*volume integrals: sum the vector values with respect to a volume (performed as a triple integral for each coordinate)
 [[Category:vector analysis]]
 [[Category:calculus]]
 [[Category:mathematics]]