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• 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

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