conservative vector fields
(1.4 hours to learn)
A vector field is conservative if it can be expressed as the gradient of a function (called the potential function), or equivalently, if all line integrals are path independent. The notion is useful because dynamical systems are easier to analyze if they can be described in terms of a potential function.
This concept has the prerequisites:
- Know the Fundamental Theorem of Calculus for line integrals
- Know what a conservative vector field is and why it is useful
- Show the equivalence of two definitions of a conservative vector field:
- can be expressed as the gradient of a function
- line integrals are independent of the path
- Know the definition of the curl of a vector field
- Be able to tell if a vector field is conservative by determining if the curl is 0
- Be able to determine the potential function for a conservative vector field
Core resources (read/watch one of the following)
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
- Section 8.3.1, "The Fundamental Theorem of Calculus for line integrals," pages 351-353
- Section 8.3.2, "Finding a potential function," pages 353-357
- differential forms
- exterior derivative
- This resource gives a more formal treatment than the others.
→ Multivariable Calculus
An introductory multivariable calculus textbook.
Location: Section 15.3, "The Fundamental Theorem and independence of path," pages 1030-1036
-No Additional Notes-
- 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