4.10 More on partial derivatives

We close the chapter with a little more concerning partial derivatives.Thus far, we have discussed functions of a single variable, either real or complex. However, it is difficult not to think of a function of one complex variable $z=x+iy$ as equally well being a function of the two real variables $x$ and $y.$ We will write $(a,b)$ and $a+bi$ to mean the same point in $C\equiv R^2,$ and we will write $\left|\right(a,b\left)\right|$ and $|a+bi|$ to indicate the same quantity, i.e., the absolute value of the complex number $a+bi\equiv (a,b).$ We have seen in [link] that the only real-valued, differentiable functions of a complex variable are the constant functions.However, this is far from the case if we consider real-valued functions of two real variables, as is indicated in [link] . Consequently, we make the following definition of differentiability of a real-valued function of two real variables.Note that it is clearly different from the definition of differentiability of a function of a single complex variable,and though the various notations for these two kinds of differentiability are clearly ambiguous, we will leave it to the context to indicate which kind we are using.

Let $f:S\to R$ be a function whose domain is a subset $S$ of $R^2,$ and let $c=(a,b)$ be a point in the interior $S^0$ of $S.$ We say that $f$ is differentiable, as a function of two real variables, at the point $(a,b)$ if there exists a pair of real numbers $L_1$ and $L_2$ and a function $\theta$ such that

$$f(a+h_1,b+h_2)-f(a,b)=L_1{h}_1+L_2{h}_2+\theta (h_1,h_2)$$

and

$$\underset{\left|\right(h_1,h_2\left)\right|\to 0}{lim}\frac{\theta (h_1,h_2)}{\left|\right(h_1,h_2\left)\right|}=0.$$

One should compare this definition with part (3) of [link] .

Each partial derivative of a function $f$ is again a real-valued function of two real variables, and so it can have partial derivatives of its own.We use simplifying notation like $f_{xyxx}$ and $f_{yyyxyy...}$ to indicate “higher order” mixed partial derivatives. For instance, $f_{xxyx}$ denotes the fourth partial derivative of $f,$ first with respect to $x,$ second with respect to $x$ again, third with respect to $y,$ and finally fourth with respect to $x.$ These higher order partial derivatives are called mixed partial derivatives.