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3blue1brown touches on that way of thinking about functions/derivatives in this video 

(very similar to Khan academy video linked in the article)


In this paragraph:


>In single variable calculus, the chain rule can be written as . In the multivariable case, for a function , we write  where  is the partial derivative of  with respect to its th argument. This can be simplified by employing the following notation, which uses a dot product.


What does  stand for? 
My understanding that g maps  and  indicates i-th element of a vector that g produces.

Is that the right way to think about it?