
We define differentiability in two dimensions as follows. A function $f$ of two variables is differentiable at argument $({x}_{0},{y}_{0})$ if the surface it defines in $(x,y,f)$ space looks like a plane for arguments near $({x}_{0},{y}_{0})$ .
(Given any positive numerical criterion, there is a circle around $({x}_{0},{y}_{0})$ within which its graph differs from the plane by less than that criterion.)
Recall that a plane in variables $f,x$ and $y$ is defined by a linear equation that can be put in the form
$f(x,y)=a(x{x}_{0})+b(y{y}_{0})+f({x}_{0},{y}_{0})$ (A)
The plane that $f$ resembles here is called the tangent plane to $f$ at $({x}_{0},{y}_{0})$ and the function it represents is called the linear approximation to $f$ defined at $({x}_{0},{y}_{0})$ .
The quantities $a$ and $b$ are called the partial derivatives of $f$ with respect to $x$ and with respect to $y$ , and written as follows
Here $a$ is the directional derivative of $f$ in the direction of the $x$ axis, $\widehat{i}$ , while $b$ is the directional derivative of $f$ in the direction of the $y$ axis, $\widehat{j}$ .
The linear approximation to $f(x,y)$ at arguments ${x}_{0}$ and ${y}_{0}$ which describes the tangent plane to $f$ at $({x}_{0},{y}_{0})$ therefore takes the form
where the vector $\stackrel{\u27f6}{\text{grad}}f$ , called the gradient vector to $f$ at $({x}_{0},{y}_{0})$ , is the vector whose components are the partial derivatives of $f$ in the $x$ and $y$ directions at the point $({x}_{0},{y}_{0})$
We generally do not write out the cumbersome subscripts that indicate the point $({x}_{0},{y}_{0})$ at which the gradient and linear approximation are defined, because they are so cumbersome, and simply write
Notice that we can write $\stackrel{\u27f6}{\text{grad}}f$ as $(\frac{\partial}{\partial x}\widehat{i}+\frac{\partial}{\partial y}\widehat{j})f$ or $\stackrel{\u27f6}{\nabla}f$ with the symbol $\stackrel{\u27f6}{\nabla}$ , called "del" representing the combination: $(\frac{\partial}{\partial x}\widehat{i}+\frac{\partial}{\partial y}\widehat{j})$ .
