3 independence of path explanations needed

[kyle_can]

Hi guys! Having trouble wrapping my head around independence of path. Trying to give an explanation to these problems. Any corrections and clarifications would be helpful and appreciated. Thanks!



my explanation: If you are for example, trying to compute energy using the fundamental theorem of calculus, path independence means that the work perform does not depend on the endpoints?





my explanation: If you are for example, trying to compute energy using the fundamental theorem of calculus, path independence means that the work perform does not depend on the endpoints?



my explanation: the line integral around any closed path is 0 and the line integral around ALL closed paths is zero, therefore leading to path independence.

 

[HallsofIvy]

Hi guys! Having trouble wrapping my head around independence of path. Trying to give an explanation to these problems. Any corrections and clarifications would be helpful and appreciated. Thanks!

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my explanation: If you are for example, trying to compute energy using the fundamental theorem of calculus, path independence means that the work perform does not depend on the endpoints?

Completely the contrary! It means that the integral depends only on the endpoints.

View attachment 4090


my explanation: If you are for example, trying to compute energy using the fundamental theorem of calculus, path independence means that the work perform does not depend on the endpoints?

Isn't this what you just said? Also neither of these two problems says anything about "energy". You should give a purely mathematical explanation. (If F'(x)= f(x) then \(\displaystyle \int_a^b f(x)dx= F(b)- F(a)\).)

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my explanation: the line integral around any closed path is 0 and the line integral around ALL closed paths is zero, therefore leading to path independence.

I think you mean IF the line integral around any closed path is 0 then we have path independence- and vice versa.

 

[kyle_can]

Thanks HallsofIvy!