Definition:Euclidean Directional Derivative

Definition

Let $\R^n$ be the $n$-dimensional real vector space.

Let $p \in \R^n$ be a point.

Let $T_p \R^n$ be tha tangent space of $\R^n$ at $p$.

Let $v \in T_p \R^n$ be a vector.

Let $\map {\mathfrak{X}} {\R^n}$ be the space of smooth vector fields of $\R^n$.

Let $Y \in \map {\mathfrak{X}} {\R^n}$ be a vector field.

The Euclidean directional derivative of $Y$ in the direction $v$ is defined by:

$\ds \bar \nabla_v Y := \sum_{j \mathop = 1}^n \map v {Y^j} \valueat {\dfrac {\partial}{\partial x^j} } p$

where

$\ds \forall i \in \N_{> 0} : i \le n : \map v {Y^i} = \sum_{j \mathop = 1}^n v^j \dfrac {\partial \map {Y^i} p}{\partial x^j}$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. The Problem of Differentiating Vector Fields