# Proving things like $d(a+b)=da+db$

This may sound a bit too elementary but at school, differential equations are taught in a haste . I am quite shaky when it comes to rigorously prove identities like $d(a+b)=da+db$.So, can anyone please point me to a source where I can learn to rigorously prove them and understand what really is going on?

Thanks!

• Do you mean linearity of differentiation? – Alex Oct 26 '12 at 6:05
• If you do mean linearity of differentiation, then consider the limit definition of a derivative. Usually by the time that is talked about, you already have the fact that if $\lim_{x\to x_0}A(x)= a$ and $\lim_{x\to x_0}B(x)= b$, then $lim_{x\to x_0}(A+B)(x)= a+b$ and similar theorems. If you wish to derive those statements, then you will need to use the epsilon-delta definition of limits; for examples about how to work with $\varepsilon$-$\delta$ arguments, you can refer to any number of online resources for intro calculus classes, and try to generalize. – Eric Stucky Oct 26 '12 at 6:10

What you want to do is prove that differential one forms behave linearly. You can attack this at several levels of depth. The most shallow way is to simply use the result that differentiation is a linear operator (which, with respect to proving $d(a+b)=da+db$, is not very far from "because I said so," but okay conceptually). To be more thorough, you can state the one form definitions concretely in terms of differentiation definitions, then use (or show) that the limits behave linearly.