Definition:Gradient Operator/Real Cartesian Space
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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:
| \(\displaystyle \grad f\) | \(:=\) | \(\displaystyle \nabla f\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}\) | $\quad$ Definition of Del Operator | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k\) | $\quad$ | $\quad$ |
In $3$ dimensions with the standard ordered basis $\left({\mathbf i, \mathbf j, \mathbf k}\right)$, this is usually rendered:
| \(\displaystyle \operatorname {grad} f\) | \(:=\) | \(\displaystyle \nabla f\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } f\) | $\quad$ Definition of Del Operator | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\partial f} {\partial x} \mathbf i + \dfrac {\partial f} {\partial y} \mathbf j + \dfrac {\partial f} {\partial z} \mathbf k\) | $\quad$ | $\quad$ |
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.
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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 operator 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:
| \(\displaystyle \nabla f\) | \(=\) | \(\displaystyle \begin{bmatrix} \frac {\partial f} {\partial x_1} \\ \frac {\partial f} {\partial x_2} \\ \vdots \\ \frac {\partial f} {\partial x_n} \end{bmatrix}\) | $\quad$ | $\quad$ |
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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.
