A line integral (also called a path integral) is the integral of a function taken over a line, or curve.
The integrated function might be a vector field or a scalar field; The value of the line integral itself is the sum of the values of the field at all points on the curve, weighted by a scalar function. That weight function is commonly the arc length of the curve, or—if you’re integrating over a vector field—the scalar product with a vector differential in the curve.
It’s this weighting which sets a line integral apart from the integrals studied early on in calculus—simpler integrals defined on intervals.
Example of a Line Integral
A simple example of a line integral is finding the mass of a wire if the wire’s density varies along its path.
If you were to divide the wire into x segments of roughly equal density (as shown above), you could sum all of the segment’s densities to find the total density using the following mass function:
Where:
- dxi = length of each segment
- λi = linear density of each segment.
However, if those line segments approach a length of zero, you could integrate to find a more accurate number for density.
Applications of Line Integrals in Physics
Many simple physical formulas can be written in terms of line integrals. For example, work is force times distance:
W = F · s.
If an object is moving through an electric or gravitational field, you can write it as:
The Line Integral in a Scalar Field
The animation below shows how a line integral over some scalar field f can be thought as the area under the curve C along a surface z = f(x,y), where z is described by the field.
If C is a smooth, piecewise curve, and f is a scalar field such that
then
Here r: [a,b]→ C is an arbitrary bijective parameterization of the cure C such that—and this is key—r(a) and r(b) give the endpoints of C. a is also defined to be less than b.
References
Fleisch, D. (2008). A student’s Guide to Maxwell’s Equations. Cambridge University Press.
Sharma, A. Text Book of Vector Calculus.
Vector Calculus: Line Integrals. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section16.02.html on March 8, 2019
Calculus Quest Study: Introduction to Line Integrals. Retrieved from https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lint/lint.html on March 8, 2019
Vector Calculus: Fundamental Theorem of Line Integrals. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section16.03.html on March 8, 2019.
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