The Directional Derivative, its Definition and Physical Interpretation
In Nut Shell: The simple derivative of a function, y, of one independent variable, x,
can be viewed as giving the derivative of y in the direction of x. This same concept
applies to functions of more than one independent variable and any given direction.
The directional derivative of a function, Duf(x,y), gives the rate of change of f(x,y) on a
line through the point, P, in the direction of the unit vector, u. In the figure below
∂/∂x represents the change of the function (slope) in the x-direction,
∂/∂y represents the change of the function (slope) in the y-direction,
and Du f represents the change of the function (slope) through P in the direction of
the unit vector u.
One can show that the directional derivative of f(x,y) in the direction of the unit vector, u
is the dot product of the gradient of f(x,y) with the unit vector, u. It provides a convenient
method to calculate the directional derivative.
Du f(x,y) = grad f(x,y) ∙ u
Recall that the gradient function, grad f(x,y), points in the direction in which the function
f(x,y) increases (or decreases) most rapidly and the dot product with u gives the
component of grad f(x,y) in the direction of u .
The maximum value of the directional derivative is in the direction of grad f(x,y). So it
occurs when u is a unit vector in the direction of grad f .
So u = grad f / | grad f |
i.e. If f (x ) represents the temperature, then proceeding in the direction of u
gives the greatest temperature change.
Copyright © 2011 Richard C. Coddington
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