Symbols:Nabla

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Symbol

Nabla is the name of the symbol $\nabla$.


The $\LaTeX$ code for \(\nabla\) is \nabla .


Del Operator

$\nabla$


Let $\mathbf V$ be a vector space of $n$ dimensions.


Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.


The del operator is a unary operator on $\mathbf V$ defined as:

$\nabla := \displaystyle \sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k}$

where $\mathbf v = \displaystyle \sum_{k \mathop = 0}^n x_k \mathbf e_k$ is an arbitrary vector of $\mathbf V$.


Laplacian

$\nabla^2$


Real-Valued Function

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

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

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

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


The Laplacian of $f$ is defined as:

\(\displaystyle \nabla^2 f\) \(:=\) \(\displaystyle \nabla \cdot \left({\nabla f}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} }\right) \cdot \left({\sum_{k \mathop = 1}^n \dfrac {\partial f} {\partial x_k} \mathbf e_k}\right)\) Definition of Del Operator
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \dfrac {\partial^2 f} {\partial {x_k}^2}\)


Vector-Valued Function

Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$.

Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$.


Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$.


The Laplacian of $\mathbf f$ is defined as:

\(\displaystyle \nabla^2 \mathbf f\) \(:=\) \(\displaystyle \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf f} {\partial {x_k}^2} }\right)\)


The $\LaTeX$ code for \(\nabla^2\) is \nabla^2 .


Also known as

The symbol nabla is also known as del, from its use for the Del operator.


Linguistic Note

The term nabla derives from the ancient Greek word νάβλα for a Phoenician harp.

This arises from the shape of the nabla symbol: $\nabla$.

The term was originally suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait.

As a result of this suggestion, the term was used in correspondence between Tait and James Clerk Maxwell, mainly in a jocular context.

The name gained official traction as a result of its adoption by Lord Kelvin in his lectures.


Sources