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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

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