where ux, uy are the unit vectors along x and y, and fx, fy are the partial derivatives of f along the x and y directions, respectively, which are given by
The gradient is associated with the concept of a directional derivative of a
function. Let us assume we have a direction u = aux + buy. The directional derivative of f at x0,y0 along u is
The gradient can be thus defined as a function of the ordered derivatives:
where the operation is the dot product of two vectors (for v = cux + duy, v 
u = ac + bd).
This expression means that the maximum value of the directional derivative as
a function of the direction u is given by the size of the gradient, and it occurs exactly when the
direction u coincides with the gradient direction.
Moreover, we can also find this direction pretty easily. Let us consider the
curve C(x,y), defined as the line in the x, y plane where the function f has a constant value (this line is called the level curve or the contour of f). At a point x0, y0 in C the rate of change of f in the direction of the unit vector u tangent to C must be zero (see the preceding definition), that is,
But this implies that the gradient vector is perpendicular to the tangent
vector u of the level curve at x0,y0. This explains the graphical construction outlined in the text.
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