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→Volume integrals: Category
== Line integrals ==
== 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.
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.
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]]
== Surface integrals==
[[Category:calculus]]
[[Category:mathematics]]
The surface integral of a vector is the summation of the [[dot product]] of the vector function with the outward unit normal perpendicular for each point on the surface. In physics this is known as the ''flux'' of the vector function over the surface region.
The surface integral of non-vector functions is also possible, as is the surface integral of the [[cross product]] of a vector function with the unit normal for every point on the surface.
Surface integrals are typically calculated by performing double integrals over two orthogonal coordinates.
=== Example ===
Calculate the surface integral of ''f(x,y)'' as the square root of ''x<sup>2</sup> + y<sup>2</sup>'' over an ''xy''-planar region ''S'' bounded by ''x<sup>2</sup> + y<sup>2</sup> = 64'':
:<math>\iint_S f(x,y)\, dx\, dy </math>
== Volume integrals==
The volume integral of a vector or non-vector function is simply the triple integral of the function over the orthogonal coordinates of the space. It is useful in physics and mechanical engineering and plays a key role in the [[Divergence Theorem]].
The volume integral is also known as the "space integral."
[[Category:Vector Analysis]]
[[Category:Calculus]]
[[Category:Mathematics]]