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I was reading in Singularities of differentiable maps by Arnold, Gusein-Zade and Varchenko and I didn't even get through the first page before being confused.

My confusion concerns the sentence about the second differential of a function. This is supposed to be a quadratic form. So in what way should I view this since this should also just work for single variable functions.

I keep going back to either d(df)=0. Somehow I must be looking for the second derivative in some form.

I guess my question is just, what exaclty is the second differential of a function (in what way is it a quadratic form)?

Cheers