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Say

##z = f(x,y)##

then I learned that

##dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy##

##dz = f_x*dx + f_y*dy##

The question is whether this expression on ##dz## is really proper. The question comes from this: I know the definition of say ##\frac{dz}{dt}## as

##\frac{dz}{dt}=lim_{\Delta t->0}\frac{z(t+\Delta t)-z(t)}{\Delta t}##

I understand ##dz/dt## as a ratio, and I can live with extending the notation a little bit to define a thingie ##dz = (something) dt##, but in my mind that only makes sense as I understand that's a shorthand for the ratio ##dz/dt##.

But how about the expression of ##dz## in terms of ##dx## and ##dy##? I know it works fine and is extremely practical (I use it to solve differential equations), and it kinda makes sense as I know I can bring it back to a ratio being a sum of two ratios, but that seems suspiciously like an abuse of notation, as it is it implies that the thingies ##dx## and ##dy## exist by themselves and form an algebra such that I can go around adding, subtracing, dividing and multiplying.

Well, I never heard of that algebra, but if there's truly an "algebra of infinitesimals", then I could write things like

##du=dx*e^{dy/dz-dw/dt}+\sqrt{dx*dy+dw*dt}##

what sounds preposterous to me (maybe it isn't). So, is there indeed an algebra on infinitesimals ##dx, dy, dz,...## or is that just a practical shorthand that only makes sense if you restrict it to linear combination forms?

##z = f(x,y)##

then I learned that

##dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy##

##dz = f_x*dx + f_y*dy##

The question is whether this expression on ##dz## is really proper. The question comes from this: I know the definition of say ##\frac{dz}{dt}## as

##\frac{dz}{dt}=lim_{\Delta t->0}\frac{z(t+\Delta t)-z(t)}{\Delta t}##

I understand ##dz/dt## as a ratio, and I can live with extending the notation a little bit to define a thingie ##dz = (something) dt##, but in my mind that only makes sense as I understand that's a shorthand for the ratio ##dz/dt##.

But how about the expression of ##dz## in terms of ##dx## and ##dy##? I know it works fine and is extremely practical (I use it to solve differential equations), and it kinda makes sense as I know I can bring it back to a ratio being a sum of two ratios, but that seems suspiciously like an abuse of notation, as it is it implies that the thingies ##dx## and ##dy## exist by themselves and form an algebra such that I can go around adding, subtracing, dividing and multiplying.

Well, I never heard of that algebra, but if there's truly an "algebra of infinitesimals", then I could write things like

##du=dx*e^{dy/dz-dw/dt}+\sqrt{dx*dy+dw*dt}##

what sounds preposterous to me (maybe it isn't). So, is there indeed an algebra on infinitesimals ##dx, dy, dz,...## or is that just a practical shorthand that only makes sense if you restrict it to linear combination forms?

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