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• ODEs
  where $AX$ is any $1$-chain from $A$ to $X$ in $K$, i.e., any $1$-chain $AX=a\in K^{(1)}$ with $\partial a=X-A$. ...h is always the case when $K$ is acyclic). Indeed, if $a'$ is another such chain, we have
  47 KB (8,415 words) - 15:46, 1 December 2015
• More ODEs
  [[image:parametric curve in tangent bundle.png| center]] ==ODEs of chain maps==
  26 KB (4,649 words) - 12:43, 7 April 2016
• Multilinear algebra
  ...espondence. Consequently, the set of all multivectors is thought of as the chain complex of the simplex: Thus, we have a chain complex:
  49 KB (8,852 words) - 00:30, 29 May 2015
• Topology Illustrated -- Index
  *[[augmented chain complex|augmented chain complex]] *[[boundary operator of the chain bundle|boundary operator of the chain bundle]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Tangent bundles
  ...e oriented edge in ${\mathbb R}$, or a combination of edges. It is a $1$-''chain''. Furthermore, the force $F$ defines $W$ as a linear function of $D$. It i Our conclusion doesn't change: $D$ is a $1$-chain and $F$ is a $1$-form. Even though this idea allows us to continue our stud
  16 KB (2,753 words) - 13:55, 16 March 2016
• Calculus on cubical complexes
  ...ference is that these numbers aren't the coefficients of the cells in some chain but the ''values'' of the $1$-cochain on these cells. The idea becomes expl ...a number to each interval to indicate the values to be integrated and the chain indicates how many times the interval will appear in the integral, typicall
  36 KB (6,218 words) - 16:26, 30 November 2015
• Discrete forms
  ...ference is that these numbers aren't the coefficients of the cells in some chain but the ''values'' of the $1$-cochain on these cells. The idea becomes expl ...a number to each interval to indicate the values to be integrated and the chain indicates how many times the interval will appear in the integral, typicall
  35 KB (6,055 words) - 13:23, 24 August 2015
• Tangent bundle
  ...f(p(t))$, so $y$ is a [[parametric curve]] in the $y$-axis. Then by the [[Chain Rule]], ==Tangent bundle==
  5 KB (859 words) - 02:33, 22 January 2013
• Forms
  ...ver a path in the complex is the ''line integral'' of $\varphi$ over a $1$-chain $a$ in $K$. It is simply the sum of the values of the form at the edges in [[image:tangent bundle dim 1.png|center]]
  13 KB (2,459 words) - 03:27, 25 June 2015
• Axioms of calculus
  *from topological spaces to a certain kind of chain complexes, $Ch(\mathcal{Mod})$ be the category of [[chain complex]]es of $R$-modules.
  6 KB (1,124 words) - 14:17, 4 August 2013
• Metric tensor
  ...epends only on the topology of the spaces, in the above sense. Indeed, the chain rule implies the following elementary result. However, the tangent bundle $T_1(K)$ is inadequate because this time we need to consider ''two directio
  42 KB (7,131 words) - 17:31, 30 November 2015
• Differential calculus of parametric curves
  [[image:tangent bundle dim 1 and 2.png| center]] ==Compositions and the Chain Rule==
  130 KB (22,842 words) - 13:52, 24 November 2018
• Metric complexes
  However, the tangent bundle $T(K)$ is inadequate because this time we need to consider ''two directions '''Definition.''' The ''tangent bundle of degree''<!--\index{tangent bundle}--> $2$ of $K$ is defined to be
  35 KB (5,871 words) - 22:43, 7 April 2016
• Geometry
  ...epends only on the topology of the spaces, in the above sense. Indeed, the chain rule implies the following elementary result. However, the tangent bundle $T_1(K)$ is inadequate because this time we need to consider ''two directio
  41 KB (6,928 words) - 17:31, 26 October 2015
• Notation
  $X,Y$ cell complexes, $M,N$ chain complexes ...X):=\{C_n(X):n=0,1,2,...\}$, or $C(X):=\oplus _n C_n(X) \quad$ the (total) chain group of $X$;
  8 KB (1,519 words) - 16:30, 1 December 2015
• Products
  <!--150-->[[image:Mobius band as a fiber bundle.png|center]] We need to understand what happens to the chain groups first.
  44 KB (7,951 words) - 02:21, 30 November 2015
• Geometry in calculus
  ...ntation]] preserving [[diffeomorphism]] of the $x$-axis and then use the [[chain rule]]. What about the discrete case? is the ''[[discrete tangent bundle]]'' defined as the disjoint union of all ''tangent spaces'' with each simpl
  9 KB (1,604 words) - 18:08, 27 August 2015
• Test
  #[[Tangent bundle]] #[[Cubical tangent bundle]]
  16 KB (2,139 words) - 23:01, 9 February 2015
• Discrete Calculus. An Introduction
  ##[[Cubical chain complex]] ##[[Tangent bundle]]
  4 KB (466 words) - 19:07, 8 July 2014
• Applied Topology and Geometry
  #[[Tangent bundle]] #[[Cubical tangent bundle]]
  16 KB (2,088 words) - 16:37, 29 November 2014

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