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• Chains vs cochains
  *they form [[vector space]]s of the same dimension -- they are [[dual spaces]]! The [[basis]] of the $k$-chains, $C_k$, is the $k$-cells, say,
  4 KB (635 words) - 18:28, 22 August 2015
• ODEs
  Recall that both are dual $0$-forms. *the velocity $v=r'$ is a dual $0$-form;
  47 KB (8,415 words) - 15:46, 1 December 2015
• Hodge duality of differential forms
  by its action on the basis elements: The dimensions of any two dual forms are complementary.
  8 KB (1,072 words) - 17:59, 24 April 2013
• Lefschetz theory for coincidences
  ...or $H_{k}(M)$ and $\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for $H^{k}(M).$
  19 KB (3,563 words) - 15:20, 9 December 2012
• Integration of differential forms of degree 0 and 1
  Note: Here, $\{1\}$ is the basis of $T{\bf R}$ ...co" in "cochain". It's a [[linear function]] to ${\bf R}$ on chains (see [[Dual spaces]]).
  12 KB (1,906 words) - 17:44, 31 December 2012
• Linear Algebra by Messer
  ...rd topics of [[linear combination]]s, [[linear independence]], [[span]], [[basis]], and [[dimension]]s are discussed. Some proofs here are too terse in my o ...n the determinants. I was pleased to see a few [[commutative diagram]]s. [[Dual space]]s are included as an option.
  2 KB (325 words) - 19:45, 7 March 2016
• Euler characteristic of graphs
  *a certain basis of the space of cycles in $G$. Next, the idea of $G$ is that of the "dual graph" of $T$:
  11 KB (1,876 words) - 19:23, 10 February 2015

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