Difference between revisions of "Vector integration"
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*ordinary integrals, indefinite or definite | *ordinary integrals, indefinite or definite | ||
| − | *line integrals | + | *line integrals: the vector values with respect to points on a line |
| − | *surface integrals | + | *surface integrals: sum the vector values with respect to an area (performed as a double integral for each coordinate) |
| − | *volume integrals | + | *volume integrals: sum the vector values with respect to a volume (performed as a triple integral for each coordinate) |
| + | == Ordinary integrals == | ||
| + | The ordinary (definite or indefinite) integral of a vector is done by integrating each orthogonal component separately. | ||
| + | |||
| + | == 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. | ||
| + | |||
| + | 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:vector analysis]] | ||
[[Category:calculus]] | [[Category:calculus]] | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 20:16, December 27, 2009
Vector integration refers to four types of integrals of vectors:
- ordinary integrals, indefinite or definite
- line integrals: the vector values with respect to points on a line
- surface integrals: sum the vector values with respect to an area (performed as a double integral for each coordinate)
- volume integrals: sum the vector values with respect to a volume (performed as a triple integral for each coordinate)
Ordinary integrals
The ordinary (definite or indefinite) integral of a vector is done by integrating each orthogonal component separately.
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.
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 P1 to P2 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.