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added section on line integrals
*ordinary integrals, indefinite or definite
*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)
== 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]]