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• Functions in higher dimensions
  *the physical space, dimension $3$, ...numbers, the graph of a function of one variable lies in the $xy$-plane, a space of dimension $2$.
  113 KB (19,680 words) - 00:08, 23 February 2019
• Products
  ...he product of a space with the segment ${\bf I}$ means “thickening” of the space. For example, the product ...oduct''<!--\index{product}--> $X \times Y$ of $X$ and $Y$ is a topological space defined on the product set $X \times Y$ with the following basis:
  44 KB (7,951 words) - 02:21, 30 November 2015
• Multilinear algebra
  ...to handle directions appears, separately, at every point of the Euclidean space. The set of all possible directions at point $A\in {\bf R}^n$ form a vector space of the same dimension. It is $V_A$, a copy of ${\bf R}^n$, attached to each
  49 KB (8,852 words) - 00:30, 29 May 2015
• Functions
  ...r 1, we visualized a sequence of position of a falling ball by “separating space and time”. We gave the former a real line and the latter a line of intege Accommodating finer and finer representations of space or time will require to continue to divide the intervals in half until it s
  151 KB (25,679 words) - 17:09, 20 February 2019
• Relative topology
  This idea applies to all topological spaces<!--\index{topological space}-->. ...--\index{topology}--> via ''neighborhoods'', a subset $A$ of a topological space $X$ with basis $\gamma$ will acquire its own collection $\gamma _A$ as the
  34 KB (6,089 words) - 03:50, 25 November 2015
• Homotopy
  ...ver, there is a profound reason ''why'' they must all have one hole. These space are homeomorphic! Informally, we say that one space can be “deformed into” the other.
  46 KB (7,846 words) - 02:47, 30 November 2015
• Metric tensor
  In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a more general setting. A module equipped with an inner product is called an ''inner product space''.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Forms
  ...need for considering directions becomes clearer when the dimension of the space is $2$ or higher. We use ''vectors''. ...The set of all possible directions at point $A\in {\bf R}^n$ form a vector space of the same dimension. It is $V_A$, a copy of ${\bf R}^n$, attached to each
  13 KB (2,459 words) - 03:27, 25 June 2015
• Calculus of sequences
  They can be used for studying the space and locations, as follows. ...more popular approach is the following. The idea is to ''separate time and space'', give a separate real line, an axis, to each moment of time, and then bri
  113 KB (18,425 words) - 13:42, 8 February 2019
• Computing homology
  ...t $\partial$ is a [[homomorphism]] (or a [[linear operator]] in the vector space case): is a linear operator between two copies of ${\bf R}^3$ with [[basis of vector space|bases]] $\{a, b, c \}$ and $\{A, B, C \}$ respectively. The values of $\par
  26 KB (4,370 words) - 21:55, 10 January 2014
• Vector fields
  As we progress in time and space, new numbers are placed in the next row of our spreadsheet. There is a ''se We continue with the rest in the same manner. As we progress in time and space, numbers and vectors are supplied and placed in each of the four sets of co
  91 KB (16,253 words) - 04:52, 9 January 2019
• Quotients
  These realizations, however, were placed within a specific Euclidean space ${\bf R}^N$. We will see that this is unnecessary. ...ays start with $X$ assumed to be a topological space<!--\index{topological space}-->, the quotient has been, so far, ''just a set''. We can't simply assume
  26 KB (4,538 words) - 23:15, 26 November 2015
• Calculus on cubical complexes
  ...d'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces (cubes). We denote it by ${\mathbb R}^n$. ...associativity, commutativity, distributivity, etc. Thus, we have a vector space:
  36 KB (6,218 words) - 16:26, 30 November 2015
• Configuration spaces
  Its space of parameters is the torus: ...space'' as the set of all positions reachable by a robot's end-effector in space;
  6 KB (921 words) - 17:14, 27 August 2015
• Set-valued maps
  ...umber of degrees of freedom of the robot, to the $3$-dimensional operating space. ...is called a ''control system''. For example, in cruise control, $M$ is the space of all possible values of the car's speed and $U$ is the engine's possible
  24 KB (4,382 words) - 15:52, 30 November 2015
• Discrete forms
  ...d'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces. We denote it by ${\mathbb R}^n$. ...associativity, commutativity, distributivity, etc. Thus, we have a vector space:
  35 KB (6,055 words) - 13:23, 24 August 2015
• Continuous functions
  '''Definition.''' Suppose we have a topological space $X$, then we can define the ''identity function''<!--\index{identity functi '''Exercise.''' Suppose $X$ is a topological space<!--\index{topological space}-->.
  42 KB (7,138 words) - 19:08, 28 November 2015
• Metric complexes
  In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a more general setting. A module equipped with an inner product is called an ''inner product space''.
  35 KB (5,871 words) - 22:43, 7 April 2016
• Applications of integral calculus
  ...shape up is the following. Suppose we have a quantity $Q$ “contained” in a space region $R$: area, volume, mass (below), etc. Then, A certain quantity, $f$, is “spread” around locations in space; for now, it is an interval within the $x$-axis. This quantity may be: leng
  103 KB (18,460 words) - 01:01, 13 February 2019
• Functions of several variables
  \text{dimension}&\text{ambient space}&\text{“hyperplane”}&\\ A hyperplane is something very “thin” relative the whole space but not as thin as, say, a curve.
  97 KB (17,654 words) - 13:59, 24 November 2018
• More on vector spaces
  ...des [[Euclidean space]]es, another important class of examples of [[vector space]]s is... ...-wise, input-wise: for each $x$, $f(x) \in {\bf R}$. ${\bf R}$ is a vector space! Use it.)
  14 KB (2,471 words) - 21:48, 5 September 2011
• Applications of ODEs
  ...o any mutual location of the rabbit and the hound as well to pursuits in a space of any dimension. ...cise.''' Implement a simulation of planetary motion in the $3$-dimensional space. Demonstrate that the motion is planar.
  50 KB (8,692 words) - 14:29, 24 November 2018
• Multilinear forms
  Given a vector space $V$, how does one ''compute'' the (algebraic) lengths, areas, volumes, etc ...of such $k$-forms over $V$ is denoted by $\Lambda ^k(V)$. It is a [[vector space]].
  18 KB (3,325 words) - 13:32, 26 August 2013
• Compact spaces
  ...nd $f(b)$. It follows from this theorem that the image of a path-connected space<!--\index{path-connectedness}--> (under a continuous map<!--\index{continuo ...will rely on the following familiar concept. A point $x$ in a topological space $X$ is called an accumulation point<!--\index{accumulation point}--> of sub
  19 KB (3,207 words) - 13:06, 29 November 2015
• Vector and complex variables
  Of course, any Euclidean space ${\bf R}^n$ can be -- in a similar manner -- rotated (around various axes), ...ppose we have addition and scalar multiplication carried out in the domain space of $A$:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Cell complex
  ...y have seen two ways to construct topological spaces<!--\index{topological space}--> from cells. Cubical sets<!--\index{cubical sets}--> are unions of cubes ...exes are built from data and its cells can then be realized in a Euclidean space.
  40 KB (6,459 words) - 23:27, 29 November 2015
• Forms in Euclidean spaces
  ...space of continuous $k$-forms is denoted by $\Omega^k({\bf R}^n)$ and the space of discrete forms is $T^k({\mathbb R}^n)$. The above argument applies to show that in $3$-space the direction variables are independent from the location variables $x$, $y
  44 KB (7,778 words) - 23:32, 24 April 2015
• Systems of ODEs
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...i.e., the collections of $(t,x(t),y(t))$ lie in the $3$-dimensional $txy$-space. That is why, we, instead, plot their ''images'', i.e., the collections of
  63 KB (10,958 words) - 14:27, 24 November 2018
• Geometry
  In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a more general setting. A module equipped with an inner product is called an ''inner product space''.
  41 KB (6,928 words) - 17:31, 26 October 2015
• The gradient
  ...the $y$-axis representing the dimensions of the input space and the output space. The first column consists of all parametric curves and the first row of al ...st dimension $3$).''' If $G$ is exact on a partition of a box in the $xyz$-space with component functions $p$, $q$, and $r$, then
  74 KB (13,039 words) - 14:05, 24 November 2018
• ODEs
  ...e continuous? The time is $K={\mathbb R}$, which seems discrete, while the space is $R={\bf R}$, which seems continuous. Let's take an alternative point of *the space is algebraic.
  47 KB (8,415 words) - 15:46, 1 December 2015
• Applications of differential calculus
  As we progress in time and space, new numbers are placed in the next row of our spreadsheet. This is how the We continue with the rest in the same manner. As we progress in time and space, a number is supplied and are placed in each of the columns of our spreadsh
  59 KB (10,063 words) - 04:59, 21 February 2019
• Algebra of differential forms
  For now, $1$-forms in the $3$-space appear to be functions of $x$, $y$, $z$, $dx$, $dy$, and $dz$ that are line First, we are given the "ambient space" which will be assumed to be Euclidean, ${\bf R}^n$.
  11 KB (1,947 words) - 18:14, 22 August 2015
• Triangulations
  ...omplex may be built as the union of a collection of subsets of a Euclidean space, while a cell complex is built via the quotient construction<!--\index{quot Representation of a topological space<!--\index{topological space}--> as a realization of a simplicial complex is also called a ''triangulati
  30 KB (5,172 words) - 21:52, 26 November 2015
• Lefschetz numbers in control theory
  *$U$ is the space of inputs, *$M$ is the space of states, and
  17 KB (3,052 words) - 22:12, 15 July 2014
• PDEs
  ...tandard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...time. Since $a$ is an $(n-1)$-cell, $k$ is an $(n-1)$-form with respect to space. It is also a $0$-form with respect to time.
  44 KB (7,469 words) - 18:12, 30 November 2015
• Integer-valued calculus
  Because there is a hole in the space, there is a vector field that is irrotational but not gradient. And vice ve ...the integral over a half of the interval? An infinite divisibility of the space implies an infinite divisibility of the ring of numbers...
  27 KB (3,824 words) - 19:07, 26 January 2019
• Several variables
  ...pter. Of course, the motion metaphor -- $x$ and $y$ are coordinates in the space -- will be superseded. In contrast to this approach, we look at the two qua ...tead of plotting all points $(t,x,y)$, which belong to the $3$-dimensional space, we just plot $(x,y)$ on the $xy$-plane -- for each $t$.
  76 KB (13,017 words) - 20:26, 23 February 2019
• The algebra of cubical cells
  We know that we can decompose the $N$-dimensional Euclidean space into blocks, the $N$-cells. For instance, this is how an object can be repr Thus, our approach to decomposition of space, in any dimension, boils down to the following:
  46 KB (7,844 words) - 12:50, 30 March 2016
• Diffusion equation
  ...tandard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...time. Since $a$ is an $(n-1)$-cell, $k$ is an $(n-1)$-form with respect to space. It is also a $0$-form with respect to time.
  35 KB (5,917 words) - 12:51, 30 June 2016
• Topology of graphs
  ...pological questions about specific spaces. Given a subset of the Euclidean space: One can also think of a graph as just a collection of points in space, also called “vertices”, or “nodes” connected by paths, called “e
  25 KB (4,214 words) - 16:08, 28 November 2015
• Differential calculus of parametric curves
  *$(f(t),g(t))$ is thought of as the position in space at time $t$. The motion may also be in the $3$-dimensional space:
  130 KB (22,842 words) - 13:52, 24 November 2018
• Simplicial complexes
  ...see it, we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ==Simplices in the Euclidean space==
  30 KB (5,021 words) - 13:42, 1 December 2015
• Simplicial homology
  ...ata'', and yet we can still study the topology of subsets of the Euclidean space -- via realizations of graphs. We will follow this latter route with simpli *A representation of a topological space $X$ as a homeomorphic image of a realization of a simplicial complex $K$ is
  31 KB (5,170 words) - 13:44, 1 December 2015
• Calculus exercises: part I
  ...unction $f$ is given below. Sketch the graph of the derivative $f′$ in the space under the graph of $f$. Identify all important points and features on the g ...unction $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
  49 KB (8,436 words) - 17:14, 8 March 2018
• Social choice
  *the space of choices is a simplicial complex $W$; '''Theorem (Impossibility).''' Suppose the space of choices $W$ is path-connected and has torsion-free homology. Then the so
  47 KB (8,030 words) - 18:48, 30 November 2015
• Homology of maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  41 KB (6,926 words) - 02:14, 21 October 2015
• Cell maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  42 KB (7,005 words) - 03:10, 30 November 2015
• Differential equations
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...placed in the first row of the spreadsheet and, as we progress in time and space, new numbers are placed in the next row of our spreadsheet:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Motion planning in robotics
  Now, the [[configuration space]] of a ''two-joint'' arm is the [[torus]]: ...e the lengths of the arms. Under the assumption $R_1>R_2$, the operational space is the annulus.
  5 KB (786 words) - 20:58, 27 August 2015
• Tangent bundles
  ...need for considering directions becomes clearer when the dimension of the space is $2$ or higher. We use ''vectors''. First, as we just saw, the work of th ...e set of all possible directions at point $A\in V={\bf R}^2$ form a vector space of the same dimension. It is $V_A$, a copy of $V$, attached to each point $
  16 KB (2,753 words) - 13:55, 16 March 2016
• The algebra of oriented cells
  ...free finitely-generated abelian group</TD> <TD>a finite-dimensional vector space over ${\bf R}$</TD> </TR> ...rt (${\bf Z}_2$ for the Klein bottle)</TD> <TD>a finite-dimensional vector space</TD> </TR>
  36 KB (6,395 words) - 14:09, 1 December 2015
• Higher order votes
  ...ncept of the ''orthogonal complement'' of a subset $P$ of an inner product space $V$: '''Proposition.''' Suppose $P$ is a subset of an inner product space $V$. Then its orthogonal complement is a summand:
  41 KB (6,942 words) - 05:04, 22 June 2016
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  The union of any collection of pixels is a subset of the [[Euclidean space|Euclidean plane]]. Therefore it acquires its topology from the plane [12] ( .... R., Harvey, R., and Cawley, G. C., “The segmentation of images via scale-space trees”, British Machine Vision Conference, 33-43 (1998).
  41 KB (6,854 words) - 15:05, 28 October 2011
• Topological spaces
  The pair $(X,\tau)$ is called a ''topological space''<!--\index{topological space}-->. The elements of $\tau$ are called ''open sets''<!--\index{open sets}-- “Open” disks on the plane, and balls in the Euclidean space, are also open.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Vector spaces: introduction
  '''Definition:''' A ''vector space'' is a set $V$, where two operations are defined: Main idea: '''A vector space is "closed" under these operations'''.
  14 KB (2,238 words) - 17:38, 5 September 2011
• Homotopy and homotopy equivalence
  ...ver, there is a profound reason ''why'' they must all have one hole. These space are homeomorphic! Informally, we say that one space can be “deformed into” the other.
  45 KB (7,738 words) - 15:18, 24 October 2015
• Inner product spaces: part 1
  Note there is no measuring in a vector space. But, in that case, there are ''no distances, no limits, no calculus''... '''Plan:''' Take a vector space and equip it with extra structure, so that we ''can'' measure.
  14 KB (2,404 words) - 15:04, 13 October 2011
• Data made Euclidean
  ...o see it we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ==Simplices in the Euclidean space==
  31 KB (5,219 words) - 15:07, 2 April 2016
• Cubical complex
  For objects located in a Euclidean space, we would like to devise a data structure that we can use to first represen Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also t
  29 KB (4,800 words) - 13:41, 1 December 2015
• Calculus of chain maps
  ...b R}$, ${\mathbb R}_x$ and ${\mathbb R}_y$, possibly representing time and space respectively. We are to study functions, that will possibly represent motion in space. They have to somehow respect the cell structure of ${\mathbb R}$. Let's re
  41 KB (7,344 words) - 12:52, 25 July 2016
• Precalculus
  One can already see how harder is to visualize things in the $3$-dimensional space, which further justifies the need for the algebraic treatment of geometry t
  100 KB (16,148 words) - 20:04, 18 January 2017
• Classes of functions
  One can already see how harder is to visualize things in the $3$-dimensional space, which further justifies the need for the algebraic treatment of geometry t ...on we take up the second geometric task, ''directions'', in the Euclidean space equipped with the Cartesian coordinate system.
  143 KB (24,052 words) - 13:11, 23 February 2019
• Pattern recognition in computer vision
  ...nsight into the nature of the problem. Once all the data is in a Euclidean space (no matter how large) all [[statistics|statistical]], [[topological data an ...k at the [[distance]] between points – images - in this 10,000-dimensional space. It can be defined in a number of ways, but as long as it is symmetric we h
  9 KB (1,526 words) - 17:54, 1 July 2011
• Cubical calculus
  ...c. These “expressions” are called ''differential forms''<!--\index{acyclic space}--> and each of them determines such a new function. That's why we further ...tter of ''calculus'', the calculus of differential forms<!--\index{acyclic space}-->:
  25 KB (4,238 words) - 02:30, 6 April 2016
• Maps of polyhedra
  Without refining the target space, repeating this approximation doesn't produce a sequence $g_n$ convergent t ...ether the set is included in one of the elements of the cover. In a metric space, it's simpler:
  51 KB (9,162 words) - 15:33, 1 December 2015
• Partial differential equations
  ...because our temperature distribution function $w$ is then a $0$-form in a space of any dimension. Recall that a partition of a ''box'' $B$ in the $txy$-space comes from partitions of its three edges as described in Chapter 20:
  53 KB (9,682 words) - 23:19, 18 November 2018
• Euler and Lefschetz numbers
  *(2) If $Y$ is a subspace of vector space $X$, then '''Fixed Point Problem.''' If $X$ is a topological space and $f:X \to X$ is a self-map, does $f$ have a fixed point: $x\in X$ such t
  41 KB (7,169 words) - 14:00, 1 December 2015
• The algebra of oriented cubes
  ...case of higher dimensions will require using the product structure of the space. ==The boundary of a cube in the $N$-dimensional space==
  32 KB (5,480 words) - 02:23, 26 March 2016
• Topology
  ...ns and coming (or not coming) back will produce information about loops in space. These loops, or $1$-cycles, are used to detect tunnels in the Universe. ...a new item: space, or, more accurately: a 3-dimensional space. How such a space creates a 3-''cycle'' may be hard or impossible to visualize. Nonetheless,
  20 KB (3,407 words) - 21:46, 30 November 2015
• Chain complexes
  If $R$ is a field, an $R$-module is a vector space. ...uld like to be able to study functions defined on subsets of the Euclidean space.
  33 KB (5,293 words) - 03:06, 31 March 2016
• Calculus exercises: part III
  ...et tall. Represent the motion as a parametric curve in the $3$-dimensional space. *Find an equation of the spiral in space converging to the origin as below (view from above):
  46 KB (8,035 words) - 13:50, 15 March 2018
• The algebra of cells
  We know that we can decompose the $n$-dimensional Euclidean space into $n$-dimensional blocks, the $n$-cells. For example, this is how an obj Thus, our approach to decomposition of space, in any dimension, boils down to the following:
  34 KB (5,644 words) - 13:35, 1 December 2015
• How to compute homology
  ...rticle we summarize the procedure for computing the [[homology as a vector space|homology]] of a [[cell complex]], by hand. ...[[chain group]] $C_k(K)$ is given as a vector space with [[basis of vector space|basis]] consisting of the cells of the complex:
  6 KB (1,049 words) - 09:21, 3 September 2011
• Stability Analysis
  ...roblem from which they were obtained, when the values of both the time and space steps are allowed to all tend to zero. Due to known results from Numerical ...OGETHER WITH OUR BOUNDARY CONDITIONS, to find expressions for the time and space components of our separated solution; using this, and a clever trick from a
  12 KB (2,051 words) - 03:51, 11 August 2012
• Diffusion
  Note: When the domain isn't the whole space, the pipes at the border of the region have to be “removed”. Here we us ...n be written simply as $Qdt$, where $Q$ is a dual $1$-form with respect to space.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Wedge product of continuous forms
  or in $3$-space Now, for the $2$-dimensional space we've got all we need. All $2$-forms are given by:
  14 KB (2,417 words) - 18:16, 22 August 2015
• Simplicial complexes and maps
  ...see it, we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ...uclidean space; for example, a plane (and a square) in the $3$-dimensional space is oriented by a choice of one of the two unit normal vectors.
  27 KB (4,625 words) - 12:52, 30 March 2016
• Algebra on graphs
  ...a+sb,\ \forall r,s \in {\bf R}$. Then $\{a,b \}$ is a basis of this vector space and the idea becomes:
  36 KB (6,177 words) - 02:47, 21 February 2016
• Calculus I, the discrete version
  ...e standard domain, the discrete representation of ${\bf R}$. Second, the ''space'' is given by ${\bf R}$, at the simplest. ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  64 KB (11,521 words) - 19:48, 22 June 2017
• Chain maps
  ...have proven the identity for all basis elements, simplices, of the vector space, $C_k(K)$, then the two linear operator coincide. $\blacksquare$
  47 KB (8,115 words) - 16:19, 20 July 2016
• Lefschetz theory for coincidences
  ...ility of control systems dictates the need for a higher dimensional domain space $N$. For example, the projection of the [[torus]] on the [[circle]] is such ...y differential equation: $M$ is the space, $F$ is the time, and $N$ is the space-time; and
  19 KB (3,563 words) - 15:20, 9 December 2012
• Introduction to discrete calculus
  Second, the ''space'' is given by any ring $R$, in general. For all the derivatives to make sen ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  40 KB (6,983 words) - 19:24, 23 July 2016
• How to compute Betti numbers
  One can acquire the [[Betti numbers]] from the [[homology as a vector space|homology groups]] (and [[cohomology]]) by taking their dimensions/ranks. Ho ...[[chain group]] $C_k(K)$ is given as a vector space with [[basis of vector space|basis]] consisting of the cells of the complex:
  5 KB (890 words) - 14:47, 24 August 2014
• Quotients of vector spaces
  Given a [[vector space]] $L$ and a subspace $M$. How do we "remove" $M$ from $L$? Unfortunately, $L \setminus M$ isn't a vector space!
  6 KB (1,115 words) - 16:03, 27 August 2015
• Cochain complexes and cohomology
  both are [[vector space]]s, very familiar objects. ...hat the set $P = \{1, x, x^2, \ldots\}$ is [[linearly independent]] in the space of functions $C({\bf R}) = \Omega^0({\bf R})$.
  17 KB (2,592 words) - 14:38, 14 April 2013
• Selection votes
  Then, instead of a single simplex, ''the space of outcomes is a simplicial complex''. The complex is meant to represent al Do we ever face a space of outcomes with a more complex topology, such as one with holes, voids, et
  24 KB (3,989 words) - 01:56, 16 May 2016
• Cubical complexes
  For objects located in a Euclidean space, we would like to devise a data structure that we can use to first represen Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also t
  20 KB (3,319 words) - 14:18, 18 February 2016
• Discrete forms and cochains
  ...ain''<!--\index{cochains}--> on $K$ is any linear function from the vector space of $k$-chains to $R$: '''Proposition.''' The $k$-cochains on complex $K$ form a vector space denoted by $C^k=C^k(K)$.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Simplicial maps and chain maps
  ...have proven the identity for all basis elements, simplices, of the vector space, $C_k(K)$, then the two linear operator coincide. ...plex}--> $K$ can be realized as a topological space<!--\index{ topological space}-->. The way to construct it is by treating the list of vertices and simpli
  34 KB (5,897 words) - 16:05, 26 October 2015
• More ODEs
  ...We also take as a model a fluid flow. The “phase space” ${\bf R}^2$ is the space of all possible locations. Then the position of a given particle is a funct ...point. Thus, there is one vector at each point picked from a whole vector space:
  26 KB (4,649 words) - 12:43, 7 April 2016
• Calculus exercises: advanced
  ...the linear map $L(x_1,x_2)=<3x_1+x_2,x_1-x_2>$ find the basis of the null space (the kernel). *(a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\b
  14 KB (2,538 words) - 18:35, 14 October 2017
• Torus
  ==As a product space== ==As a quotient space and a cell complex==
  5 KB (708 words) - 22:46, 3 September 2011
• Means
  ...he group may have no division. If these are points in a subset of a vector space, the set may be non-convex. ...--\index{algebraic mean}--> if it is a homomorphism. For $X$ a topological space, a mean is ''topological''<!--\index{topological mean}--> if it is a contin
  10 KB (1,914 words) - 03:05, 6 November 2018
• Simplicial maps
  ...e have proven the identity for all basis elements, simplices of the vector space, $C_k(K)$, then the two linear operator coincide. ...mplex}--> $K$ can be realized as a topological space<!--\index{topological space}-->. The way to construct it is by treating the list of vertices and simpli
  34 KB (5,929 words) - 03:31, 29 November 2015
• Vector calculus: review
  #(a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\b #Suppose that a mass $M$ is fixed at the origin in space. When a particle of unit mass is placed at the point $(x,y)$ other than the
  7 KB (1,394 words) - 02:36, 22 August 2011
• Freshman's introduction to discrete calculus
  ...e standard domain, the discrete representation of ${\bf R}$. Second, the ''space'' is given by ${\bf R}$, at the simplest. ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  42 KB (7,443 words) - 14:18, 1 August 2016
• Combinatorial cell maps
  If $R$ is a field, an $R$-module is a vector space. ...complex of time. What is the other chain complex $C$, the chain complex of space? Since these two forms take their values in ring $R$, we can choose $C$ to
  31 KB (5,330 words) - 22:14, 14 March 2016
• Integration
  '''Example (space shift).''' If $y$ is the location and we change the place from which we sta
  69 KB (11,727 words) - 03:34, 30 January 2019
• Inner product
  ...gles between them. An inner product is how one adds geometry to a [[vector space]]. Given a vector space $V$, an ''inner product'' on $V$ is a function that associates a number to
  4 KB (749 words) - 20:12, 1 May 2013

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