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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

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