(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The variation:

deltau = (6x^2y+3lny)dx + (2x^3+3x/y)dy

is an exact differential and u is given by

u = 2x^3y+3xlny

Demonstrate by integrating deltau along the two different paths from point A(1,1) to point B(3,2) (one path has a constant y from x=1 to x=3 and then the x is constant from y=1 to y=2, the second path is just the opposite, first the x is held constant until y=2 and then y is constant until x=3, essentially each path is two line segments that make up a rectangular looking graph) that the integral is path independent and equal to u(B)-u(A).

2. Relevant equations

3. The attempt at a solution

So far I have tried differentiate the equation twice. The first time setting y=0, x=2, dy=0 then doing it again with y=1, x=0, and dx=0. For the x=0 integration I got 10 and for y=0 I got 42.5 and then I added them together. I then substituted in y=x^2. That is my second path, however as I am writing this out I am realizing this is probably very wrong. Regardless I did the same thing integrating with the substituted y as I did earlier. This gave an answer of 341.5. My conclusion is that 52.5 does not equal 341.5 and since this is an exact differential both answers should be the same due to the path independence....

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Integration along two separate paths.

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