The Directional Derivative, its Definition
and Physical Interpretation
∂/∂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. |
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One can show that the directional derivative of f(x,y) in the direction of the unit vector, u, Du f(x,y) , 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.
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 . |
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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. Click here for more
discussion of the directional derivative. |
Copyright © 2017 Richard C. Coddington
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