# R2->R2 Calculus

• domhal
In summary, the conversation discusses the concept of differentiation of functions from R2 to R2, which is briefly mentioned in a complex analysis course. The book considers an example function f(x,y) = (x,-y) and explains that its derivative at a point is a linear map represented by its Jacobian matrix. The derivative of a function is a linear function that best approximates the function at that point. It is recommended to read the Bartle Elements of Real Analysis for a more thorough understanding of differentiation of such functions.

#### 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

$$\lim_{h \rightarrow 0} \left ( f(\mathbf{x + h}) - f(\mathbf{x}) = \mathbf{h} Df(x) \right )$$
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.