Category:Gradient
This category contains results about Gradient.
Definitions specific to this category can be found in Definitions/Gradient.
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 $f$ 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 $\ds \mathbf x = x_1 \mathbf e_1 + x_2 \mathbf e_2 + \cdots + x_n \mathbf e_n = \sum_{k \mathop = 1}^n x_k \mathbf e_k$ be a vector in $\R^n$.
Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$.
The gradient of $f$ (at $\mathbf x$) is defined as:
| \(\ds \grad f\) | \(:=\) | \(\ds \nabla f\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf x}\) | Definition of Del Operator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf x} } {\partial x_k} \mathbf e_k\) |
This category currently contains no pages or media.