(1.3 hours to learn)
The line integral gives a notion of integrating a vector field along a curve. Common uses include computing work done by a force and finding potential functions corresponding to a gradient field.
This concept has the prerequisites:
- functions of several variables
- vector fields (Vector fields are something we compute line integrals of.)
- Be able to compute line integrals of vector fields with respect to curves
- Be able to express the line integral in terms of the arc length parameterization, or in terms of an arbitrary parameterization
- Why does the line integral of a vector field depend on the orientation of a curve?
- Be aware that the line integrals are independent of the parameterization of a path, for a given orientation
Core resources (read/watch one of the following)
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
→ Multivariable Calculus
An introductory multivariable calculus textbook.
Location: Section 15.2, "Line integrals," pages 1019-1028
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 8.3, "Line integrals and Green's Theorem," not counting subsections, pages 348-351
- differential forms
- exterior derivative
- This gives a more rigorous treatment than the other resources.
- Green's Theorem is a powerful theorem relating line integrals to integrals of functions
- Surface integrals are the higher-dimensional analogue
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