When can I apply the idea of differentials?

[LucasGB]

My calculus book says sometimes derivatives can be regarded as the ratio of differentials, and sometimes they can't. Apparently, there's a similar rule for integrals. When can I think of derivatives and integrals as operations with differentials? And when can't I?

 

[HallsofIvy]

My calculus book says sometimes derivatives can be regarded as the ratio of differentials, and sometimes they can't. Apparently, there's a similar rule for integrals. When can I think of derivatives and integrals as operations with differentials? And when can't I?

What exactly does your book say? What are its exact words. While, strictly speaking, derivatives are NOT ratios, I can't think of a case in which they cannot be regarded as ratios of differentials.

 

[wofsy]

My calculus book says sometimes derivatives can be regarded as the ratio of differentials, and sometimes they can't. Apparently, there's a similar rule for integrals. When can I think of derivatives and integrals as operations with differentials? And when can't I?

The slope of a tangent line can be thought of as the deviation of y, dy, for a given deviation of x, dx. This is a ratio.

 

[LucasGB]

What exactly does your book say? What are its exact words. While, strictly speaking, derivatives are NOT ratios, I can't think of a case in which they cannot be regarded as ratios of differentials.

Actually, now that I think about it, aren't all derivatives the ratio of differentials? I say this based on the definition of differentials:

dx = (dx/dy)dy, where (dx/dy) is the derivative of x with respect to dy. Therefore:

dx/dy = (dx/dy). What do you think?

PS. 1: I think this whole differential business is one big mess because every author seems to be afraid to address the subject in a clear way. The consequence is that students like me have to go through great lengths in order to try and understand whether differentials are, or aren't, a legitimate concept of mathematics which can be employed without fear.

PS. 2: Thank you all for your replies.

 

[wofsy]

Actually, now that I think about it, aren't all derivatives the ratio of differentials? I say this based on the definition of differentials:

dx = (dx/dy)dy, where (dx/dy) is the derivative of x with respect to dy. Therefore:

dx/dy = (dx/dy). What do you think?

PS. 1: I think this whole differential business is one big mess because every author seems to be afraid to address the subject in a clear way. The consequence is that students like me have to go through great lengths in order to try and understand whether differentials are, or aren't, a legitimate concept of mathematics which can be employed without fear.

PS. 2: Thank you all for your replies.

originally differentials were considered to be small deviations of the variables and their ratio was an approximation to the derivative. The question was - given a very small deviation in one variable what is the deviation on the other provided these deviations are small - where small means small enough to give a good linear approximation - much like a regression.

So for instance if x^2 - y =2 then at the point (1,1) 2dx - dy = 0 meaning twice the small deviation in x is the small deviation in y, approximately - where "approximately" means that this gives the best linear relationship for small enough deviations. I always think of differentials as small deviations and in Physics that is how they are thought of.

In modern mathematics differentials are thought of as linear 1 forms that map tangent vectors into tangent vectors. This is a formalism that obscures this fundamental intuition.