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• Functions in higher dimensions
  From this point of view, a ''vector'' is a pair, $PQ$, of locations $P$ and $Q$. ...d directions and angles between them. Our interest here is the ''algebraic operations'' on vectors.
  113 KB (19,680 words) - 00:08, 23 February 2019
• The gradient
  ...tion of two variables in the two main directions. The result is given by a vector called its gradient. ...ndence, we attach this vector to the point it came from? The result is a ''vector field''. It is a function from ${\bf R}^2$ to ${\bf R}^2$ and it is placed
  74 KB (13,039 words) - 14:05, 24 November 2018
• Differential calculus of parametric curves
  *The single dependent variable is ''multi-dimensional'', a point or a vector in ${\bf R}^n$. As we know, the former point, $X$, is the end of the latter vector, $OX$. In either case, this is just a combination of two function of the sa
  130 KB (22,842 words) - 13:52, 24 November 2018
• Vector and complex variables
  We transition to from numbers to ''vectors''. But what are the operations? This is a $2$-dimensional vector.
  113 KB (18,750 words) - 02:33, 10 December 2018
• More on vector spaces
  Besides [[Euclidean space]]es, another important class of examples of [[vector space]]s is... Observation: There are algebraic operations on functions $f \colon {\bf R} \rightarrow {\bf R}$:
  14 KB (2,471 words) - 21:48, 5 September 2011
• Vector spaces: introduction
  ...important, algebraically! The point is that we can perform some algebraic operations with these entities if we define them properly. ...'t use this notation here. From context, it should be clear when this is a vector.
  14 KB (2,238 words) - 17:38, 5 September 2011
• Transformations
  We are to solve a ''vector equation''; i.e., to find these unknown coefficients: ...way to stretch these two vectors so that the resulting combination is the vector on the right:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Multilinear algebra
  *$1$-vector $v_1 \in V$; *$2$-vector $v_1 \wedge v_2$ with $v_1,v_2 \in V$;
  49 KB (8,852 words) - 00:30, 29 May 2015
• Matrices as functions
  ==Vector functions== ...se are [[vector space]]s, ${\bf R}^2$. We just combined $u$ and $v$ in one vector $(u,v)$.
  13 KB (2,187 words) - 22:17, 9 September 2011
• Functions of several variables
  \text{number}& &&&\text{point or vector} \text{point or vector}& &&&\text{number}
  97 KB (17,654 words) - 13:59, 24 November 2018
• Forms vs vector fields and functions
  ==Vector fields vs discrete 1-forms== Let's recall that a [[vector field]] is given if there is a vector attached to each point of the plane:
  8 KB (1,153 words) - 19:42, 17 April 2013
• Parametric curves
  [[Image:motion along a vector.jpg|right]] #$f(t) = v \cdot t$, a motion along a [[vector]] $v \in {\bf R}^n$ (constant speed), then $f(0) = 0, f(1) = 1$.
  34 KB (5,665 words) - 15:12, 13 November 2012
• Vector functions
  We have considered two types of "vector functions": ...too consider the general vector functions, i.e., both input and output are vector:
  28 KB (4,769 words) - 19:42, 18 August 2011
• Identities of vector calculus
  ...d $V,W$ are [[vector field]]s. Again, the ideas for the arrows come from [[vector calculus]]: *[[gradient]]: functions $\mapsto$ vector fields,
  6 KB (879 words) - 13:00, 17 April 2013
• Continuous functions
  ...terpret the proposition in terms of $C[a,b]$ (assume $A:=B:=[a,b]$). These operations make $C[a,b]$ into?... ...and only if it is continuous with respect to each of its variables. And a vector-valued function is continuous if and only if every of its coordinate funct
  42 KB (7,138 words) - 19:08, 28 November 2015
• Quotients of vector spaces
  Given a [[vector space]] $L$ and a subspace $M$. How do we "remove" $M$ from $L$? Unfortunately, $L \setminus M$ isn't a vector space!
  6 KB (1,115 words) - 16:03, 27 August 2015
• Linear functions in Euclidean space
  '''A linear function preserves vector operations:''' '''Vector functions.''' What if we have a [[vector function]]
  23 KB (3,893 words) - 04:43, 15 February 2013
• Homology groups of graphs
  ...k requests. If the joints are ordered, these requests can be recorded in a vector format, coordinate-wise. For example, $(1,0,1,0,…)$ means: flip the first Recall that a function that satisfies this identity “preserves the operations” of the group and such functions are called ''homomorphisms''. The extens
  28 KB (4,685 words) - 17:25, 28 November 2015
• Calculus Illustrated
  15 The arithmetic operations on functions 1 Operations on sets
  16 KB (1,933 words) - 19:50, 28 June 2021
• Discrete forms
  ...know that the discrete differential forms, as cochains, are organized into vector spaces, one for each degree. Let's review this first. If $p,q$ are two forms of the same degree $k$, it is easy to define algebraic operations on them.
  35 KB (6,055 words) - 13:23, 24 August 2015

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