Lecture Notes: DG Categories and Homotopy Category of a DG Category

Math Questions: Differentiability, DIfferentials, Derivatives and... Derivable?

Reginald Anderson

17 February 2018

Math Questions

(I'm even just talking about real differentiability here, for instance)

Derivatives send functions to functions.

Differentials send differential (n)-forms to differential (n+1)-forms.

Want to define: differential equation

Differential equations relate functions of higher derivatives to a function. Where is the “differential” in differential equations? In slope fields, and graphing? In statements about boundaries, divergences, and curls? Where is d◦d=0 here? That if you integrate over the boundary of a boundary, you don’t?

Why is a function called differentiable at the point x, when all of its partial derivatives exist in some open neighborhood U, of x? Shouldn’t this function be called derivable, instead? Or is this statement that the function can be made into a 1-form, by applying the differential.

Attempt at an answer: Differentiable means there’s an exact 1 form which integrates to said function. [I am assuming that integration takes a differential n-form as input and spits out an n-1 form.] (Yeah, I know the limit definition of differentiability. Yeah, I know the derivative is a linear operator that satisfies the Chain Rule [also the Leibniz rule], the best local linear approximation, etc.) The function is not closed unless its derivative is zero, in which case it is called constant. But this suggests that not every function satisfies d◦d=0, because not every function differentiates to a 1 form which is constant from the point of view of a 2-form... It should be the case that if you apply the differential twice, you get 0, and I’m not sure how that occurs here. (I think this is a statement about the de Rham complex, for example.)

Thanks,

One Confused Student.