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• Algebra of sets
  ...orall A\in\mathcal{A}[A^C\in\mathcal{A}]}}<ref group="Note">Recall {{M|1=A^C:=X-A}} - the [[complement]] of {{M|A}} in {{M|X}}</ref> * <math>A\in R\implies A^c\in R</math>
  3 KB (507 words) - 18:43, 1 April 2016
• Sigma-algebra
  # {{M|X\in\mathcal{A} }} as {{M|\emptyset^C\in\mathcal{A} }} :: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this show
  8 KB (1,306 words) - 01:49, 19 March 2016
• Pre-measure
  *** {{M|1=[a,b)^c=(-\infty,a)\cup[b,+\infty)}} *** {{M|1=[a,d)-[b,c)}} where {{M|1=a< b< c< d}} is {{M|1=[a,b)\cup[c,d)}}
  5 KB (782 words) - 01:49, 26 July 2015
• Measure
  {{Requires proof|Trivial|easy=Yes|grade=C}}
  6 KB (941 words) - 14:39, 16 August 2016
• Complement
  ...{M|X}} the complement of {{M|A}} (often denoted {{M|A^c}}, {{M|A'}} or {{M|C(A)}}) is given by: <math>A^c=\{x\in X|x\notin A\}=X-A</math>
  726 B (145 words) - 13:28, 18 March 2015
• Ring
  | <math>\forall a,b,c\in R[(a+b)+c=a+(b+c)]</math> | Now writing {{M|a+b+c}} isn't ambiguous
  7 KB (1,248 words) - 05:02, 16 October 2016
• Measurable map
  * {{M|g:(B,\mathcal{B})\rightarrow(C,\mathcal{C})}} is measurable * {{M|g\circ f:(A,\mathcal{A})\rightarrow(C,\mathcal{C})}} is measurable.
  5 KB (792 words) - 02:31, 3 August 2015
• Curve
  Given a <math>f:\mathbb{R}^n\rightarrow\mathbb{R}</math> and a {{M|c\in\mathbb{R} }} we define the level curve as follows<ref> <math>\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}</math>
  1 KB (224 words) - 21:30, 28 March 2015
• Tangent space
  <math>\omega\in T_p(\mathbb{R}^n)\iff \omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} </math> is a [[Derivation|derivat ...ghtarrow\mathbb{R}</math> which satisfies the [[Leibniz rule]]. Recall {{M|C^\infty(M)}} is the set of all [[Smooth function|smooth functions]] on our [
  6 KB (1,190 words) - 19:27, 14 April 2015
• Set of all derivations of a germ at a point
  ...ref> denotes the set of all [[Derivation|derivations]] of the form <math>D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}</math>, that is - it is a deriv
  772 B (132 words) - 21:49, 13 April 2015
• Leibniz rule
  Take: <math>D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}</math> - a [[Derivation]] if it
  723 B (123 words) - 00:57, 5 April 2015
• Set of all derivations at a point
  .../math> to denote this set (the set of derivations of the form <math>\omega:C^\infty\rightarrow\mathbb{R}</math>)<ref>John M. Lee - Introduction to smoot ...Derivation at a point|derivations (at a point)]] of [[Smooth|smooth or {{M|C^\infty}}]] functions from {{M|A}} at a point {{M|p}} (assume {{M|1=A=\mathb
  2 KB (291 words) - 21:51, 13 April 2015
• Germ
  ...int {{M|p\in\mathbb{R}^n}}, we define an equivalence relation on the <math>C^\infty</math> functions defined in some neighbourhood of {{M|p}} as: * <math>f:U\rightarrow\mathbb{R}</math> which is <math>C^\infty</math> ([[Smooth|smooth]])
  2 KB (285 words) - 01:36, 5 April 2015
• The set of all germs of smooth functions at a point
  ...the special case of smooth functions, that will be assumed here (so <math>C^k_p(\mathbb{R}^n)</math> is a notation for this set, but there is no genera ...>" in the absence of a space. That is, assume {{M|C^\infty_p}} denotes {{M|C^\infty_p(\mathbb{R}^n)}}
  794 B (140 words) - 01:50, 5 April 2015
• Derivation
  If {{M|a\in\mathbb{R}^n}}, we say that a map, {{M|\alpha:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} }} is a '''''derivation at {{M|a} * Given {{M|f,g\in C^\infty(\mathbb{R}^n)}} we have:
  2 KB (325 words) - 18:08, 14 October 2015
• Smooth
  ...Second Edition</ref> <math>F:U\rightarrow V</math> is '''smooth''', <math>C^\infty</math> or ''infinitely differentiable'' if:
  870 B (148 words) - 06:38, 7 April 2015
• Smooth manifold
  * {{M|C^\infty}} manifold structure
  3 KB (413 words) - 21:09, 12 April 2015
• Smooth structure
  * {{M|C^\infty}} structure
  2 KB (246 words) - 07:10, 7 April 2015
• Motivation for tangent space
  Take the line {{M|1=y=mx+c}}, where {{M|m}} is the gradient and {{M|c}} is the intercept with the y axis, writing this we see the line can be giv | {{M|1=y=mt+c}}
  6 KB (975 words) - 00:18, 11 April 2015
• Polar equation of a line
  * {{M|1=y=mx+c}} * <math>r=\sqrt{t^2(m^2+1)+2mtc+c^2}</math>
  1 KB (223 words) - 22:43, 10 April 2015