Nabla

In differential geometry, population (also called) of) is a vector differential operator represented by the symbol: ${displaystyle nabla }$ (blah).

In three-dimensional Cartesian coordinates, nabla can be written as:

${displaystyle nabla ={hat {x}}{partial over partial x}+{hat {y}{partial over partial y}+{hat {z}}{partial over partial z}. !$

being ${displaystyle {hat {x}}}$, ${displaystyle {hat {y}}}$ and ${displaystyle {hat {z}}}$ the unit vectors in the directions of the coordinated axles. This base is also represented by ${displaystyle {hat {imath}}}$, ${displaystyle {hat {jmath}}}}$, ${displaystyle {hat {k}}}$.

Symbolism

The name of the ∇ symbol comes from the Greek word equivalent to the Hebrew word harp, a musical instrument that has a similar shape. There are related words in the Aramaic and Hebrew languages. The symbol was first used by William Rowan Hamilton, but laterally: ◁. Another lesser-known name for the symbol is atled (delta spelled backwards), because nabla is an inverted Greek letter delta (Δ): in modern Greek it is called ανάδελτα (anádelta), which means "inverted delta".

In HTML it is written ∇ and in LaTeX as nabla. In Unicode, it is the character U+2207, or 8711 in decimal notation.

Expressions of the nabla operator

Expression in non-Cartesian coordinate systems

When coordinate systems other than Cartesian coordinates are used, the nabla expression must be generalized. In orthogonal coordinate systems, such as Cartesian, cylindrical and spherical coordinate systems, scale factors appear in the expression:

${displaystyle nabla ={frac {{hat {q}}}{h_{1}}}}{partial over partial q_{1} +{frac {{hat {q}}{2}{h}{h}{h}{h }{h }{h }{f}{f}{f}{f}{f}{f}{f}}}}{f}}{f}}}{f}{f}{f}}{f}{f}}}}}{f}{f}}{f}{f}}{f}{f}}}}{f}}}}{f}}}}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}}}}}{f}}}}}{f}{f}}}}{$

In particular, for cylindrical coordinates (${displaystyle h_{rho }=h_{z}=1, h_{varphi }=rho }$)

${displaystyle nabla ={hat {rho }}{frac {partial }{partial rho }}}+{frac {hat {varphi }}{rho }}{frac {partial }{partial }{partial }{partial }{$

and for spherical coordinates (${displaystyle h_{r}=1, h_{theta }=r, h_{varphi }=r{rm {sen}}theta }$)

${displaystyle nabla ={hat {r}{frac {partial }{partial }}}}{frac {frac {theta }{frac}{partial }{partial }{partial theta }}{frac {varphi }{$

Alternative definitions

Intrinsic definition

A definition of the nabla operator can be given that does not depend on the coordinate system used. This definition is a generalization of the one used to define divergence:

${displaystyle nabla star A=lim _{Delta Vto 0}{frac {1}{Delta V}}{oint Astar d{vec {S}}}}}}}}{delta V}}}{oint Astar d{vec {$

In the previous expression ${displaystyle star }$ represents an arbitrary product (scalar, vectorial, tensorial or a scaler) and ${displaystyle A}$ it is a climbing field, vectorial or tensorial. ${displaystyle Delta V}$ is a differential volume that in the limit is reduced to a point. In this way the gradient, divergence, rotational and other operators can be intrinsically defined.

Relationship with the covariant derivative

The nabla operator also applies to differential varieties.

Given a differentiable variety provided with a connection that gives rise to a covariant derivative, the nabla operator is defined as the application of the set of functions on the variety or 0-forms to the set of 1-forms of said variety. Given a local coordinate system, it is expressed as:

${displaystyle nabla f=sum _{alpha }f_{;alpha }dx^{alpha }}}$

The covariant derivative raises the order of the tensor to which it is applied. For example for a vector field ${displaystyle {vec {v}}}$ in three dimensions its covariant derivative would be a second order tensor of 9 components (a matrix 3×3)

The covariant derivative can be represented in this context as ${displaystyle nabla otimes {vec {v}}}$Where ${displaystyle otimes }$ represents the diadic product.

With this in the face of small displacements the vector would change according to: ${displaystyle delta {vec {v}}=(nabla otimes {vec {v}}})cdot delta {vec {r}}}}}}$

Relationship with the external differential

All expressions that involve the nabla operator of vector calculation ${displaystyle mathbb {R} ^{3}$ can be expressed in terms of external differential of a n-form n ≤3 ${displaystyle mathbb {R} ^{3}$:

• The gradient of a function is associated with the outer differential of a 0-form.
• The rotation of a vector field is associated with the outer differential of a 1-form.
• The divergence of a vector field is associated with the outer differential of a 2-form.

A function is a 0-form over Euclidean space, its gradient is:

${displaystyle nabla f=(df)^{sharp },qquad (nabla f)^{i}=g^{ij}{frac {partial f}{partial x^{j}}}}}}}}}$

where ${displaystyle g^{ij}}$ are the components of the inverse of the metric tensor in the coordinates ${displaystyle {x^{j}}}$Obviously at Cartesian coordinates ${displaystyle g^{ij}=delta _{i}^{j}}$.

The rotational of a vector field can be associated with the exterior differential in a 1-way.

${displaystyle nabla times mathbf {A} =*(d(mathbf {A} ^{flat })),qquad (nabla times mathbf {A}^{i}{i}=epsilon {ijk}{frac {partial }{partial x{j}{$

where ${displaystyle}$ is the dual operator of Hodge and ${displaystyle g_{ij}}$ are the metric tensor components in the coordinates ${displaystyle {x^{j}}}$.

The divergence of a vector field can be associated with the external differential of a 2-form.

${displaystyle nabla cdot mathbf {A} =*d(*(mathbf {A} ^{flat }),qquad (nabla cdot mathbf {A}{i}{frac {1}{2}{2}{2}{epsilon ^{mjk}{frac}{partial}{$

The Laplacian of a function can be associated with the application of two exterior differentials alternating with two dual Hodge operations:

${displaystyle Delta f=*d(*df),qquad Delta mathbf {A} =*d(*dmathbf {A} ^{flat })}}$

Nabla operator applications

This operator can be applied to scalar fields (Φ) or to vector fields F, giving:

• Gradient:	${displaystyle nabla phi }$
• Divergence:	${displaystyle nabla cdot {vec {F}}}}$
• Rotation:	${displaystyle nabla times {vec {F}}}$
• Laplaciano:	${displaystyle nabla ^{2}phi =Delta phi }$

Algebra of the ∇ operator

As it is a differential operator, the result of its application on a product follows similar rules to the derivative of a product. However, depending on the character of the entities on which it acts, the result may have a more or less complicated expression. The most important formulas are:

${displaystyle nabla (phi psi)=(nabla phi)psi +phi (nabla psi}$

${displaystyle nabla cdot (phi {vec {A}})=(nabla phi)cdot {vec {A}}+phi (nabla cdot {vec {A}}}})}$

${displaystyle nabla times (phi {vec {A}})=(nabla phi)times {vec {A}}}+phi (nabla times {vec {A}}})}}$

${cHFFFFFF}{cH00FF}{cdot {vec {B}}={vec {B}}}{times {cHFFFFFFFF}{cH00FFFFFF}{B}{B}{cHFFFFFF}{cHFFFF}{cHFFFFFF}{cHFFFFFFFFFFFF}{cHFFFFFFFFFF}{cH00}{B}{cH00}{B}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{c}{cH00}{cH00}$

${displaystyle nabla cdot ({vec {A}}}times {vec {B}}})=(nabla times {vec {A})cdot {vec {B}}-(nabla times {vec {B}}})cdot {vec {A}}}}}$

${cHFFFFFF}{cH00FF}{cH00FFFF}{cH00FFFF}}{cH00FFFF}{cH00FFFF00}{cH00FFFF00}{cH00FF00}{cH00FF00}{cH00FFFF00}{cH00FFFFFF00}{cH00}{cH00FFFFFF00}{cH00FFFFFF00}{cH00}{cH00FF00FFFFFFFFFF00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00FFFFFF00}{cH00}{cH00}{cH00}{cH00FF00}{cH00}{cH00FFFFFFFFFFFFFFFF00}{cH00}{cH00FF00}{cH00FF00FF00FFFFFFFFFFFFFFFFFFFFFFFFFF$

ADVANCED BIBLIOGRAPHY

• Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5