Hi everyone, I'm new here.

I'm an italian student who has an exam in applied thermodynamics soon. Through the whole course as well as the Physics one I have faced a lot of equations expressed with differentials, basically all of them. I have never been taught how to handle them though, all I was told is that a differential (for example dU) is a very small change in the value of U and even though I have studied derivatives of functions (multivariable functions too), integrals and double integrals, I don't feel like it's enough to fully understand the meaning of such expressions.

Let's take a concrete example: my notes say that the equation for the first law of thermodynamics is

and that T, p and MU_i are the partial derivatives of the internal energy U with respect to S,V and N_i respectively and as such they are also function of the same variables S,V and Ni. It makes sense because if I partial derivate that expression with respect to the variables S,V and N_i I obtain T p and MU_i respectively.

Now, the doubts I have are

1) How is the differential form of a function obtained? Is it simply the sum of the partial derivatives with respect to each variable multiplied by the differential of the variable?

I mean

If so, I can kind of grasp the meaning of it, since it basically means multiplying the change in the variable times the rate of change and sum the contributions of all the variables; but what are the advantages of using such an equation instead of the equation of the function itself?

2) If I wanted to calculate the actual change in the internal energy (dU) how would I proceed? Integrate that expression with respect to which variable? Am I supposed to integrate it like this:

But if so, what is the meaning of this operation? I mean, summing the integrals of different parts of a function with respect to different variables has no meaning in my book. Also, if I try this operation with, say, the function Z=x*y, I get

which isn't correct.

I hope I've been clear enough, thanks in advance.