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| More generically, take the dot product of ''u⃗'' and the [[Calculus/Gradients|gradient]]. | More generically, take the dot product of ''u⃗'' and the [[Calculus/Gradient|gradient]]. |
Directional Derivative
A directional derivative is a generalization of partial derivatives.
Contents
Description
A partial derivative describes the rate of change in one variable given that all others are held constant. Concretely, the partial derivative of f(x,y) with respect to x is the rate of change in x while holding y constant.
A directional derivative expresses the rate of change in all variables in a given direction. The given direction can be expressed as angle θ, but really represents a unit vector like u⃗ = cosθi + sinθj. The dot product therefore projects the rates of change in the direction of this unit vector.
In the above bivariate example, the directional derivative is given by:
Du f(x,y) = fxcosθ + fysinθ
where fx is the partial derivative of f with respect to x and fy is with respect to y.
More generically, take the dot product of u⃗ and the gradient.