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:<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math>
:<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math>
The multiple variables in a line integral pose difficulties in solving it. Three common techniques are available:
*[[parameterization]], which reduces multiple variables to only one (typically "t"), first for the underlying curve and then for the vector function
*for a closed curve (a loop), [[Green's Theorem]] converts the line integral around the lop into a double integral over the region inside
*for a [[conservative field]], which are path-independent, find the exact differentials at the end points and calculate the difference
''Example'': The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:
''Example'': The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:
A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the curve. Put another way, the line integral is the area under the function and along the curve. The familiar, basic integral is simply the line-integral using the x-axis as the curve.
A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the curve. Put another way, the line integral is the area under the function and along the curve. The familiar, basic integral is simply the line-integral using the x-axis as the curve.
[[Category:vector analysis]]
[[Category:Vector Analysis]]
[[Category:calculus]]
[[Category:Calculus]]
[[Category:mathematics]]
[[Category:Mathematics]]