## 6.5 The Tangent Plane and the Gradient Vector

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, **i**, while b is the directional derivative of f in the direction
of the y axis, **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 **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 **grad** f as
with the symbol ,
called "del" representing the combination.