Show that F is conservative assuming it's values depend only on endpoints.

  • Thread starter Thread starter phantomcow2
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on proving that a vector field \( F \) is conservative if the line integral \( \int_C F \cdot dr \) depends solely on the endpoints of the curve \( C \). The proof relies on the definition of a conservative function, which varies among textbooks. Some texts define a conservative function as one where the integral is independent of the path taken, depending only on the endpoints. The user seeks additional resources or formal names for this proof to enhance their understanding.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically line integrals.
  • Familiarity with the definition of conservative vector fields.
  • Knowledge of parametrization of curves in \( \mathbb{R}^3 \).
  • Basic proficiency in LaTeX for mathematical notation.
NEXT STEPS
  • Research the formal definitions of conservative vector fields in various calculus textbooks.
  • Study the proof of the Fundamental Theorem of Line Integrals.
  • Explore examples of conservative fields and their properties in vector calculus.
  • Learn about the relationship between conservative fields and potential functions.
USEFUL FOR

Students in calculus courses, particularly those studying vector calculus, educators seeking to clarify concepts of conservative fields, and anyone looking to deepen their understanding of line integrals and their applications.

phantomcow2
Messages
52
Reaction score
0
Assuming that the value of [tex]\int\[/tex]F [tex]\bullet[/tex]dr, then F is a conservative function.

The class is Calc 3. My professor went through this proof in class but it was the one proof in this section that I didn't fully comprehend. Usually I try to find the proof from another source, such as online, to solidify my understanding. I'm unable to find this proof, though.

Can anybody link me to where this proof is recited, or even if it has a formal name?
Thanks.
 
Physics news on Phys.org
PS. Sorry for the crappy latex formatting. There should be a "C" underneath the integral sign, and that's supposed to be the vector valued function F dotted with dr, the parametrization of curve C. Thank you.
 
The proof depends stongly on the precise definition of "conservative". Some texts use "the integral depends only on the endpoints" as the definition of "conservative" in which case there is nothing to prove! What definition does your textbook use?
 

Similar threads

F is integrable if and only if its positive and negative parts are
  • · Replies 2 ·
Replies
2
Views
3K
Multivar; Path length, and average y-value over it
  • · Replies 2 ·
Replies
2
Views
4K
  • Sticky
How to use the "w" in www
  • · Replies 0 ·
Replies
0
Views
5K
Find the value of the line integral ?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Need to find the spring constant to achieve Max Velocity
  • · Replies 6 ·
Replies
6
Views
2K
Challenge Math Challenge - February 2019
  • · Replies 86 ·
3
Replies
86
Views
24K
Graduate How do we know the mass of an electron?
  • · Replies 28 ·
Replies
28
Views
5K
Graduate Probability via Expectation and Callen's criterion
  • · Replies 14 ·
Replies
14
Views
3K
C - getting filenames from stdin/stdout for use in getopt
  • · Replies 3 ·
Replies
3
Views
6K
Graduate How can I numerically solve for f(q) using only its derivative, f'(x)?
  • · Replies 2 ·
Replies
2
Views
2K