Total vs partial integration

1. Oct 31, 2012

superg33k

Hi,

Is there a difference between

$$\int f(x,y(x)) dx$$

And

$$\int f(x,y(x)) \partial x$$

???

If so, how is the total integral written in terms of partial integrals?

2. Oct 31, 2012

HallsofIvy

I have never seen $\partial$ used in that way.

3. Oct 31, 2012

Klungo

In both cases, the function f is dependent on x only.

If by partial integration, you mean an iterated integral, then the result of both should be the same.

Either way, the iterated integral uses a total differential dx, not a partial.

4. Oct 31, 2012

lurflurf

The dx usually means partial integration. The ∂x is an added reminder of partial integration it is sometimes used when solving exact differential equation as a reminder. Writing y(x) is also a clear indicator of functional dependence, more clear than writing y.

5. Oct 31, 2012

haruspex

Neither have I, and as other posts here point out x is the only independent variable in the OP, so it cannot make any difference.
More generally (when there's another independent variable), it could make sense as a path integral, i.e. along a path where the other independent variable is constant.

6. Oct 31, 2012

lurflurf

Have none of you read the CRC Handbook of Chemistry and Physics?

7. Nov 1, 2012

Mute

I've never seen $\partial$ used that way either (I haven't throughly read the CRC handbook, it seems), but if I had to wager a guess I would suppose that

$$\int f(x,y(x))\partial x$$
is meant to be integrated in only the first argument, holding y=y(x) fixed, while

$$\int f(x,y(x)) dx$$
is meant to be integrated over all of the x-dependence.

But, without some more context, I could be entirely wrong here.

8. Nov 1, 2012

superg33k

Thanks for your help everyone. My question has been answered above and beyond.