In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the rank of a
differentiable map between
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s at a point
is the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of
at
. Recall that the derivative of
at
is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
:
from the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at ''p'' to the tangent space at ''f''(''p''). As a linear map between
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s it has a well-defined rank, which is just the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
in ''T''
''f''(''p'')''N'':
:
Constant rank maps
A differentiable map ''f'' : ''M'' → ''N'' is said to have constant rank if the rank of ''f'' is the same for all ''p'' in ''M''. Constant rank maps have a number of nice properties and are an important concept in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
.
Three special cases of constant rank maps occur. A constant rank map ''f'' : ''M'' → ''N'' is
*an
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
if rank ''f'' = dim ''M'' (i.e. the derivative is everywhere
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
),
*a
submersion if rank ''f'' = dim ''N'' (i.e. the derivative is everywhere
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
),
*a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Form ...
if rank ''f'' = dim ''M'' = dim ''N'' (i.e. the derivative is everywhere
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
).
The map ''f'' itself need not be injective, surjective, or bijective for these conditions to hold, only the behavior of the derivative is important. For example, there are injective maps which are not immersions and immersions which are not injections. However, if ''f'' : ''M'' → ''N'' is a smooth map of constant rank then
* if ''f'' is injective it is an immersion,
* if ''f'' is surjective it is a submersion,
* if ''f'' is bijective it is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
.
Constant rank maps have a nice description in terms of
local coordinates
Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples:
* Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
. Suppose ''M'' and ''N'' are smooth manifolds of dimensions ''m'' and ''n'' respectively, and ''f'' : ''M'' → ''N'' is a smooth map with constant rank ''k''. Then for all ''p'' in ''M'' there exist coordinates (''x''
1, ..., ''x''
''m'') centered at ''p'' and coordinates (''y''
1, ..., ''y''
''n'') centered at ''f''(''p'') such that ''f'' is given by
:
in these coordinates.
Examples
Maps whose rank is generically maximal, but drops at certain singular points, occur frequently in
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s. For example, in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, the rank of the map from the two angles to a point on the sphere (formally, a map ''T''
2 → ''S''
2 from the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
to the sphere) is 2 at regular points, but is only 1 at the north and south poles (
zenith
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
and
nadir
The nadir (, ; ar, نظير, naẓīr, counterpart) is the direction pointing directly ''below'' a particular location; that is, it is one of two vertical directions at a specified location, orthogonal to a horizontal flat surface.
The direc ...
).
A subtler example occurs in
charts on SO(3)
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot ...
, the
rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
nautical engineering
Marine engineering is the engineering of boats, ships, submarines, and any other marine vessel. Here it is also taken to include the engineering of other ocean systems and structures – referred to in certain academic and professional circl ...
, and
aerospace engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is si ...
, among many other uses. Topologically, SO(3) is the
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
RP
3, and it is often desirable to represent rotations by a set of three numbers, known as
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189†...
(in numerous variants), both because this is conceptually simple, and because one can build a combination of three
gimbal
A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''
3 of three angles to the real projective space RP
3 of rotations, but this map does not have rank 3 at all points (formally because it cannot be a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
, as the only (non-trivial) covering space is the hypersphere ''S''
3), and the phenomenon of the rank dropping to 2 at certain points is referred to in engineering as ''
gimbal lock
Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
.''
References
*{{cite book , first = John , last = Lee , year = 2003 , title = Introduction to Smooth Manifolds , series = Graduate Texts in Mathematics 218 , location = New York , publisher = Springer , isbn = 978-0-387-95495-0
Smooth functions