I'm not sure how to explain this, but I have a gap in my understanding of infinitesimals/differentials. I've so far had calc 1 and 2, and have been taught that dy/dx represents a slope, which represents a change. So, my current understanding is if I had a function dy/dx(x) and plugged in x, I could deduce how much of a change in y I could expect, if I moved a unit x, at that particular x of interest.
Where my understanding falls apart is, I don't see how dx in a slope, which represents change, would also represent "change" in dA=dx*dy. I mean in the classical meaning of the word change. I know we could say "well in the same way delta X is a change in x and represents length", but I don't view it as such.
Not sure if this is related, but here's another thing bothering me, say we have P=dF/dA. Why dF and not only F? I understand dA would represent this infinitesimal area, but what does dF represent then? It's supposed to be a change in F, right? Why do we need a change?
I'd really appreciate someone clearing this up or me, because I'm expected to pass a fluid mechanics test while our uni doesn't even offer a calc 3 course. You can understand my frustration.
Thanks
Well, if you take any function $f$ which admits derivative, then you may write: $$f'(x)\qquad\text{or}\qquad\frac{df}{dx}(x)=\frac{d}{dx}f(x)$$ which are considered equivalent in notation. I guess you have been explained to take a variation $\Delta f$ and $\Delta x$, whose ratio represents a line which passes through the plot of $f$ in (at least) $2$ points. Then you think of derivation as making that line a tangent line on a point $x_{0}$. With 2 variables, say $x$ and $y$, the idea is somewhat the same. If $f(x,y)$ is now a real function in two variables, the plot will be in $\mathbb{R}^{3}$ thus it is a sort of surface. In this new setting, the idea of tangent line sounds appealing but not good enough: if you imagine a sphere and a point on its surface, then you realize there are many tangent lines through that point: but only 1 tangent plane. A variation $\Delta x\Delta y$ is now a variation in the area, so we may write $\Delta A=\Delta x\Delta y$. Passing to infinitesimal variations, we would have $d A=dxdy$. Note this little discussion lacks of mathematical rigorousness and it is meant only to help you visualize what's going on.
In terms of notation, $P=\frac{dF}{dA}$ can be written as $P=\frac{d}{dA}F$ if it pleases you more.