Line integral

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A line integral of a vector function f along a line or curve segment C is the summation of the values it takes along the curve. It can be expressed in any of these three equivalent ways:

\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}
\int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz
\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle

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:

\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}

Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then this is a conservative field and its line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero. For a conservative field, the difference in potential between the endpoints of a curve equals the line integral along that same 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.

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