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Higher Dimensions

I want to extend the treatment to we just went through to higher dimensions. In my view, this provides a very nice way of thinking about why the generalisation of the derivative to higher dimensions is what it is.  It has certainly taken me a while to appreciate the definition of the derivative of a a function of many variables and I am attempting to share some of my thoughts.

In the last post we thought about graphs of functions and tangent-lines. Now, the graph of a function is intially something 0ne thinks of a picture of that function. In my opinion, there is a sense in which one ought to continue to do so. It’s just that now, we’ll be thinking about $\mathbb{R}^m$-valued functions f defined on  an open subset U of $\mathbb{R}^n$. Recall that the graph of f is the set of points $(x, y)$ in $U \times \mathbb{R}^m$ such that f(x) = y.

What might we mean by the derivative of f ?

In these posts I will not be explaining all of the basic mathematical concepts which are required to appreciate the main discussion, so they are not aimed at the novice, as it were. I would guess maybe that a second-year university student in mathematics will be able to fully appreciate these posts.

Recently, I have been thinking about differentiation and in this post I would like to discuss some ways of approaching the concept, starting right from the basics. In this first post, we’ll discuss some approaches to the idea of differentiation. I want to settle on a certain geometric perspective and then generalise it to higher dimensions. The aim is to shed light on why the definition of the derivative of a function generalises as it does.