Vector integration

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.