Definition:Directional Derivative

Definition

Let:

$f: \R^n \to \R, \mathbf x \mapsto \map f {\mathbf x}$

be a real-valued function such that the gradient $\nabla \map f {\mathbf x}$ exists.

Let $\mathbf u$ be a unit vector in $\R^n$.

The directional derivative of $f$ in the direction of $\mathbf u$ is defined as:

$\dfrac \partial {\partial \mathbf u} \map f {\mathbf x} = \nabla \map f {\mathbf x} \cdot \mathbf u$

where $\cdot$ denotes the dot product.

Also see

• Definition:Geometrical Representation of Gradient Operator

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $2 a$. The Operation $\nabla S$
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 13.6$