Total vs Partial Integration: What Are the Key Differences?

[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?

Thanks for your help.

 

[HallsofIvy]

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

 

[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.

 

[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.

 

[haruspex]

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

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.

 

[lurflurf]

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

 

[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.

 

[superg33k]

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