Because there are infinitely many directions, there are infinitely many possible directional derivatives: the rate of change of f with distance depends on direction. We might, however, suspect that directional derivatives in three orthogonal directions are sufficient to determine the directional derivative in any direction. To show that our suspicion is well founded, assume that f can be expanded in a Taylor series about x = (x, y, z):
df df df f(x + sQx,y + sQy,z + sQz) = f(x,y,z) + sQx—- + sQy— + sQz—+0(s2), (6.7)
dx dy dz where the symbol O(s2) indicates all those terms in the series in powers of 2 or greater in s. Now divide this equation by s and take the limit as s approaches zero to obtain df df df df — — q — + n — + n —
ds dx dy dz
This is the scalar (or dot) product of U with the gradient of f :
As we suspected, to find the directional derivative in any direction, we need only the partial derivatives of f along three orthogonal axes.
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