(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:
- 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)
→ 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.
- create concept: shift + click on graph
- change concept title: shift + click on existing concept
- link together concepts: shift + click drag from one concept to another
- remove concept from graph: click on concept then press delete/backspace
- add associated content to concept: click the small circle that appears on the node when hovering over it
- other actions: use the icons in the upper right corner to optimize the graph placement, preview the graph, or download a json representation