What is the Derivative of Functions from R2->R2 and How is it a Linear Map?

[domhal]

I would like to learn about differentiation of functions from R2->R2. Such functions are mentioned briefly at the beginning of the text for my complex analysis course for sake of comparison. However, I find that I don't know very much about them.

The book considers f(x,y) = (x, -y), saying that f is differentiable and that "Its derivative at a point is the linear map given by its Jacobian...". I don't understand, for example, how the derivative is a linear map. I associate (perhaps incorrectly) the derivative with a rate of change. Could someone recommend some resources (books, websites etc) for understanding the differentiation of such functions?

Domhal

 

[MalleusScientiarum]

Think about this definition of the derivative, implicitly:
$$\lim_{h \rightarrow 0} \left ( f(\mathbf{x + h}) - f(\mathbf{x}) = \mathbf{h} Df(x) \right )$$
In this case, if $$\mathbf{h} \in R^2$$ and $$f(\mathbf{x}) :R^2 \rightarrow R^2$$ then, for the dimensionality to make sense, $$Df(\mathbf{x})$$ has to be a 2x2 matrix.

I strongly recommend the Bartle Elements of Real Analysis to read more on this. It's chapters on many variable functions are quite interesting and thorough, as is the rest of the book.

 

[HallsofIvy]

A more general definition of the derivative of any function (at a point) is the linear function that best approximates the function at that point.

In that sense, the derivative of, say, y= x², at x= 1 is not the number 2 but the linear function y= 2x.

The derivative of f(x,y)= (x,-y) is the linear function whose matrix (in the i, j basis) is the matrix with columns (1,0), (0, -1).