How do I set up integrals for differentials?

[mewmew]

Ok, I have having problems with the folloing:
http://www.physics.uc.edu/~simpson/pics/Desktop-Images/0.jpg
How exactly do I do this? I thought all I did, for example with the first du, was set up 2 integrals, one for the dx part, and one for the dy part. I then thought for (i) the dx integral would go from (a->x) and the dy integral would go from (b->y), then for (ii) I would integrate the dx from (b->y) and the dy from (a->x). To be honest though I am really pretty lost and any advice would be much appreciated.

 

[HallsofIvy]

You can't "set up 2 integrals, one for the dx part, and one for the dy part." because each contains the other variable.

How about doing what you were told to do: (i) First integrate along the line from (a,b) to (x,b) then from (x,b) to (x,y). (The use of (x,y) as a specific point is confusing. I'm going to call that point (u,v)) On the first line, take x= t(u-a)+ a, y= b with 0<t< 1. Then dx= (u-a)dt and dy= 0. Substitute those values into the two differentials and integrate. On the line from (u,b) to (u,v) let x= u and y= t(v-b)+ b. Then dx= 0, dy= (v-b)dt, 0< t< 1. Again, put those into the two differentials and integrate. Add the results to find the integrals on the entire path.

Now, do the same thing on the lines (a,b) to (a, v) to (u,v): let x= a, y= (v-b)t+ b so that dx= 0, dy= (v-b)dt, 0< t< 1, then let x= (u-a)t+ a, y= v so that dx= (u-a)dt, dy= 0, 0< t< 1.

You should have learned that integrals such as these are independent of the path if and only if the differentials are exact. For which of these do you get the same integral over both paths? (Of course, that doesn't prove that the integral will be the same over any path. That's why the problem says "could be".)

Do you remember the "cross derivative" test for exactness? If not, look it up!

 

[kyle8921]

Hello there,

It seems like you are having trouble with understanding how to set up integrals for differentials. To start, it's important to understand that differentials are infinitesimal changes in a variable, such as dx or dy. In order to integrate them, we need to set up the proper limits and determine which variable is being integrated with respect to.

In the image you provided, there are two differentials, dx and dy, and two integrals, one for each. The first thing you need to do is to determine which variable is being integrated with respect to. In this case, it looks like dx is being integrated with respect to x, and dy is being integrated with respect to y.

For the first part, you are correct in setting up the dx integral from (a to x) and the dy integral from (b to y). This means that we are integrating dx as x changes from a to x, and dy as y changes from b to y. This will give us the total change in the function from (a,b) to (x,y).

For the second part, you are also correct in setting up the dx integral from (b to y) and the dy integral from (a to x). This means that we are now integrating dx as x changes from b to y, and dy as y changes from a to x. This will give us the total change in the function from (b,a) to (y,x).

I would also recommend reviewing the concept of differentials and integrals to gain a better understanding of how they work together. You can also try practicing with simpler examples to get a better grasp on the concept.

I hope this helps and please don't hesitate to ask for further clarification if needed. Good luck!