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

  • Thread starter phantomcow2
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In summary, There is a proof for the statement "Assuming that the value of \int\F \bulletdr, then F is a conservative function" which the OP is unable to find and is asking for a link or the formal name of the proof. The proof may depend on the definition of "conservative" used in the textbook. Some definitions may not require a proof.
  • #1
phantomcow2
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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.
 
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  • #2
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.
 
  • #3
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?
 

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

What does it mean for a function to be conservative?

A function is considered conservative if its value does not change along a given path. This means that the total amount of work done by the function is independent of the path taken to get from one point to another.

How do you know if a function is conservative?

A function is conservative if it satisfies the condition that its derivative with respect to each variable is equal to the derivative of the function with respect to the other variables. This is known as the gradient condition.

What is the significance of a function being dependent only on endpoints?

If a function's values depend only on its endpoints, it means that the function is path-independent. This is a necessary condition for a function to be considered conservative.

How can you prove that a function is conservative?

To prove that a function is conservative, you must show that it satisfies the gradient condition. This can be done by taking the partial derivative of the function with respect to each variable and equating it to the derivative of the function with respect to the other variables.

Why is it important to determine if a function is conservative?

It is important to determine if a function is conservative because it allows us to make certain predictions and calculations about the behavior of the function. For example, if a force is conservative, we can use concepts such as potential energy and work to analyze and understand its effects on a system.

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