




The vector whose x,y and z components are the respective partial derivatives of f at (x, y, z), is called the gradient of f, and is written either as grad f orf. Hererepresents the "differential operator" vector,
The gradient vector points normal to the tangent plane of
f in two dimensions, and normal to the tangent hyperplane in higher
dimensions. Since it is easily computed, the tangent hyperplane is easily found
as well.
Suppose now that f is a differentiable function at (x, y, z) and we want to
find its directional derivative on a line L(d) having unit vector d
pointing along it. We can always write d as the sum of its projection into
the tangent plane of f at (x, y, z) and its projection normal to that
plane. f experiences no change at all if we move an infinitesimal distance from
(x, y, z) in the tangent plane. Only the component of d normal
to the tangent plane, and hence in the direction of the gradient, produces
change in f. The directional derivative of f in the d direction is therefore
given by the magnitude of the gradient of f there, times the component of n
normal to the tangent plane: in other words, by dgrad
f, that is, by df.
The gradient therefore gives us a quick way to compute directional derivatives.
The Fundamental Theorem of Calculus on L(d):
Since the line L(d) is one dimensional, all the results of ordinary calculus apply to f / L(d), including the fundamental theorem, which statesor with dl = dds,