Definition:Gradient Operator/Real Cartesian Space
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Definition
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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k\) |
In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:
| \(\ds \grad f\) | \(:=\) | \(\ds \nabla f\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } f\) | Definition of Del Operator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\partial f} {\partial x} \mathbf i + \dfrac {\partial f} {\partial y} \mathbf j + \dfrac {\partial f} {\partial z} \mathbf k\) |
for a vector $\mathbf u = x \mathbf i + y \mathbf j + z \mathbf k$.
On a Region
In particular: Space of functions. Notation is already defined iirc.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Let $S \subseteq \R^n$.
Let $\sqbrk {X \to Y}$ be the space of functions from $X$ to $Y$.
Suppose that for all $\mathbf x \in S$, $\map {\nabla f} {\mathbf x}$ exists.
The gradient can then be defined as an operation acting on $f$:
- $\nabla: \mathbf F \to \sqbrk {S \to \R^n}$
- $\paren {f: \mathbf x \mapsto \map f {\mathbf x} } \mapsto \paren {\nabla f: \mathbf x \mapsto \map {\nabla f} {\mathbf x} }$
where:
- $\mathbf F = \set {f \in \sqbrk {S \to \R}: \nabla f \text{ is defined} }$
That is:
| \(\ds \nabla f\) | \(=\) | \(\ds \begin{bmatrix} \frac {\partial f} {\partial x_1} \\ \frac {\partial f} {\partial x_2} \\ \vdots \\ \frac {\partial f} {\partial x_n} \end{bmatrix}\) |
If you are fairly certain that there are none, please remove {{proofread}} from the code.
Also see
- Results about gradient can be found here.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: The Gradient: $22.29$
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 13.6$
- For a video presentation of the contents of this page, visit the Khan Academy.
- WARNING: This link is broken. Amend the page to use
{{KhanAcademySecure}}and check that it links to the appropriate page.
- WARNING: This link is broken. Amend the page to use