Definition:Gradient

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Definition

Gradient of Straight Line

Let $\LL$ be a straight line embedded in a Cartesian plane.

The slope of $\LL$ is defined as the tangent of the angle that $\LL$ makes with the $x$-axis.


Gradient Operator

Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

Let $\mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \displaystyle \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$.

Let the partial derivative of $f$ with respect to $u_k$ exist for all $u_k$.


The gradient of $f$ (at $\mathbf u$) is defined as:

\(\ds \grad f\) \(:=\) \(\ds \nabla f\)
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}\) Definition of Del Operator
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k\)