nLab nabla

Idea

The operator nabla (also called Atled or Del, and written $\nabla$) is an operator introduced in quaternionic analysis and now heavily used in elementary vector analysis?. The gradient, curl, and divergence may all be defined in terms of it.

Definition

Informally, nabla is defined to be a vector (to be thought of as a tangent vector on some manifold, which is often taken to be the cartesian space $\mathbb{R}^3$) of differential operators

$$\nabla = \sum_{k = 1}^n e_k \frac{\partial}{\partial x^k} ,$$

and it may be applied to a scalar field or vector field in any way that it may be possible to multiply a scalar or vector (respectively) by a vector. Classically, this gives three applications: the gradient of a scalar field (corresponding to scalar multiplication), the divergence of a vector field (corresponding to the dot product), and the curl or rotation of a vector field (corresponding to the cross product).

Formally, all uses may be combined into a single definition based on integration on hypersurfaces. If $M$ is a smooth Riemannian manifold, $V$ and $W$ are vector bundles over $M$, $\odot$ is a smooth bilinear operator from the tangent bundle and $V$ to $W$, and $T$ is a smooth section of $V$, then $\nabla \odot T$ is a smooth section of $W$ whose value at any point $x$ is given by the integral formula

$$(\nabla \odot T)(x) = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} \odot T \,d S,\,\,\,\,\stackrel{\circ}{D}\ni x,$$

where $D$ runs over the regions with smooth boundary $\partial{D}$ containing the point $x$ and $\vec{n}$ is the unit vector of outer normal to the surface $S=\partial D$.

The formula does not depend (when everything in smooth) on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart containing balls $D = D_r$ with decreasing radius $r \gt 0$ in this particular coordinate chart. The formula also does not depend on orientation; $\partial{D}$ is naturally a pseudo-oriented? submanifold and so $\vec{n} \odot T \,d S$ (also written $d\vec{S} \odot T$) is a $W$-valued pseudo-$(n-1)$-form, and we can integrate pseudoforms on pseudo-oriented submanifolds. Perhaps most surprisingly, although it makes no sense in general to integrate $W$-valued (pseudo)forms (unless $W$ is a trivial bundle), we may simply perform the integration using any coordinate chart as if we were in the cartesian space $\mathbb{R}^n$ (where every vector bundle is trivial), and the limit is again independent of this choice.

This formula can also be extended to the case where $T$, or anything else up to $M$ itself, is only $C^k$ and not smooth, at the risk that $\nabla \odot T$ might not be defined or that the choice of coordinate chart might make a difference even in the limit. In that case, we consider that $\nabla \odot T$ is defined only if the choice of coordinate chart in fact makes no difference.

Historical and terminological notes

The nabla is an old instrument in the shape of inverted triangle, sort of an Assyrian harp. William Hamilton introduced the operator $\nabla$ (sometimes called Hamilton’s or Hamiltonian operator, but distinguish strictly from Hamiltonian in the mechanics sense) and James Maxwell used it to write his equations.

• History of nabla nabla.txt

The terms ‘Atled’ and ‘Del’ are based on the idea that the nabla is an upside-down capital Greek Delta. Sometimes the partial derivative symbol $\partial$ is taken to be the lowercase equivalent, as it is a variant (but not upside-down) of the lowercase Delta.

Related concepts

• nabla

• gradient

  • gradient flow

  • symplectic gradient

• Hamiltonian flow

• curl

• divergence

References

• G.E. Shilov, Mathematical analysis (functions of several real variables), part 1–2