Multivariable Calculus Decent Direction Proof

I am reading the proof of the following lemma:

Let be differentiable at . If , then is a decent direction for at .

I don't understand the first line of the proof, which states: because is differentiable at then

where

They define a decent direction as:

I don't understand what is exactly. Is it just the Fréchet derivative?

Re: Multivariable Calculus Decent Direction Proof

Re: Multivariable Calculus Decent Direction Proof

Chiro: I don't see what that has to do with the question.

- Hollywood

Re: Multivariable Calculus Decent Direction Proof

To me this looks like a multi-variate taylor series expansion in which one is approximating it through a linearization.

Its like how we use taylor series to expand f(x+a) for some arbitrary x or a with one of them known and the other is considered to be small. In the above case, we are doing it with a multivariable function using a vector instead of a scalar.

The Big-O notation gives us how the error behaves in regard to its parameters (namely lambda and the d vector). When these approach zero in magnitude and/or length then you get the result discussed above.

Big-O notation is used a lot in mathematics to look at how residuals or errors behave under certain conditions.