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• Vector fields
  ...o each edge representing the strength of the flow (in the direction of one of the axes). Such a system may look like this: Here the strength of the flow is shown as the thickness of the arrow. This is a real-valued $1$-form.
  91 KB (16,253 words) - 04:52, 9 January 2019
• Functions in higher dimensions
  These are the two main sources of a multiple ''dimension'': *multiple spaces of single dimension interconnected via functional relations.
  113 KB (19,680 words) - 00:08, 23 February 2019
• Vector and complex variables
  Matrices appear in systems of linear equations. ...much of each do you need to have $6$ pounds of blend with the total price of $\$14$?
  113 KB (18,750 words) - 02:33, 10 December 2018
• Classes of functions
  ...er, we will study some specific functions as well as some broad categories of functions. We start with the former. ...efer basketball? Then the “preference function” $F$ cannot be simpler: all of its values are equal and all arrows point at the basketball.
  143 KB (24,052 words) - 13:11, 23 February 2019
• Several variables
  Let's review what we learned in Chapter 7 about motion of a ball (or a cannonball). ...''vertical'' velocity is constantly changed by the gravity. The dependence of the height on the time is quadratic:
  76 KB (13,017 words) - 20:26, 23 February 2019
• Calculus of sequences
  The idea of calculus is presented in a single picture: The two problems are solved with the help of these two versions of the same elementary school formula:
  113 KB (18,425 words) - 13:42, 8 February 2019
• The integral
  We know that the ''area of a circle'' of radius $r$ is supposed to be $A = \pi r^{2}$. .... We confirm the formula with nothing but a spreadsheet. We plot the graph of the function:
  66 KB (11,473 words) - 21:36, 19 January 2019
• Manifolds
  ==What is the topology of the physical Universe?== ...sun deviates from a straight line may be considered as evidence in support of this idea:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Products
  ...to the image of a stack, which is a simple arrangement of multiple copies of $X$: More complex outcomes result from attaching to every point of $X$ a copy of $Y$:
  44 KB (7,951 words) - 02:21, 30 November 2015
• Forms in Euclidean spaces
  The simplest example of a differential form is a $1$-form over the real line: where $f$ is a function of $x\in {\bf R}$ multiplied by the second variable called $dx\in {\bf R}$.
  44 KB (7,778 words) - 23:32, 24 April 2015
• Integration
  ==Linear change of variables in integral== ...$I$ that satisfies $F' = f(x)$ for all $x$ is called an ''antiderivative'' of $f$.
  69 KB (11,727 words) - 03:34, 30 January 2019
• Sequences and their limits
  ==Limits of sequences: long-term trends== ...result is an ever-expanding string, a sequence, of numbers. If the frames of the video are combined into one image, it will look something like this:
  64 KB (10,809 words) - 02:11, 23 February 2019
• Calculus of chain maps
  Suppose we have two copies of the complex ${\mathbb R}$, ${\mathbb R}_x$ and ${\mathbb R}_y$, possibly re ...represent motion in space. They have to somehow respect the cell structure of ${\mathbb R}$. Let's recall how ''cell functions'' are introduced.
  41 KB (7,344 words) - 12:52, 25 July 2016
• Introduction to discrete calculus
  We next examine the combined ''domain'' of these new functions. We are to make the usual domain of functions -- the reals ${\bf R}$ -- discrete. We divide this set into unit
  40 KB (6,983 words) - 19:24, 23 July 2016
• Calculus on cubical complexes
  ...uantities to be studied are typically ''real numbers''. We choose our ring of coefficients to be $R={\bf R}$. ...tion, we will use the calculus terminology: ''differential forms'' instead of cochains.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Discrete forms
  ...lus, the quantities are typically the ''real numbers''. We choose the ring of coefficients to be $R={\bf R}$. ..., we will use the calculus terminology: the ''differential forms'' instead of cochains.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Freshman's introduction to discrete calculus
  ...${\bf R}$ into discrete pieces. To begin with, we divide it into intervals of equal length $h>0$: The result is two types of pieces:
  42 KB (7,443 words) - 14:18, 1 August 2016
• Homotopy
  ...quivalence relation that produces the same result for a much broader class of spaces: ...try to understand the actual mathematics behind these words, with the help of this juxtaposition:
  46 KB (7,846 words) - 02:47, 30 November 2015
• Discrete forms and cochains
  ...the definite integral over $I$ is often thought of as a function the input of which is any integrable ''function'' $f$ while the output is a real number. ...iemann integral is introduced in calculus as the limit of the Riemann sums of $f$. The student then discovers that this function is ''linear'':
  34 KB (5,619 words) - 16:00, 30 November 2015
• The algebra of oriented cells
  ==Are chains just “combinations” of cells?== The nature of the problem is topological because the ''rubber'' band is allowed to stretc
  36 KB (6,395 words) - 14:09, 1 December 2015
• Discrete calculus article
  {{Short description|Discrete analog of calculus}} {{About|discrete version of calculus|discrete exterior calculus|Discrete exterior calculus}}
  27 KB (4,329 words) - 16:02, 1 September 2019
• Calculus Illustrated
  This is a page of ''Calculus Illustrated'' by [[Peter Saveliev]], a textbook for undergraduat ...AABs388J02Zuoly8clE4WVRpa?dl Fall 2020 lectures]. They use the actual text of the book.
  16 KB (1,933 words) - 19:50, 28 June 2021
• Topology Illustrated -- Index
  *[[Arrow's Impossibility Theorem|Arrow's Impossibility Theorem]] *[[Arzela-Ascoli Theorem|Arzela-Ascoli Theorem]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Test
  <TR> <TD><center>[[Stokes theorem]]</center></TD></TR> <TR> <TD><center>[[Image:Stokes theorem deconstructed.png|center]]</center></TD></TR>
  16 KB (2,139 words) - 23:01, 9 February 2015
• Cubical calculus
  ...al of integration fixed -- is often thought of as a real-valued ''function of the integrand''. This idea is revealed in the usual function notation: ...iemann integral is introduced in calculus as the limit of the Riemann sums of $f$. The student then discovers that this function is ''linear'':
  25 KB (4,238 words) - 02:30, 6 April 2016
• Applied Topology and Geometry
  <TR> <TD><center>[[Stokes theorem]]</center></TD></TR> <TR> <TD><center>[[Image:Stokes theorem deconstructed.png|center]]</center></TD></TR>
  16 KB (2,088 words) - 16:37, 29 November 2014
• Cubical complex
  ...}^N$. Suppose also that we have its decomposition ${\mathbb R}^N$, a list of all (closed) cubical cells in our grid. ...boundary operator is well defined. This requires us to include all “faces” of the cells already present.
  29 KB (4,800 words) - 13:41, 1 December 2015
• Chain complexes
  We now review what it takes to have ''arbitrary ring of coefficients'' $R$. ...and results can be found in the standard literature such as Hungerford, ''Algebra'' (Chapter IV).
  33 KB (5,293 words) - 03:06, 31 March 2016
• Homotopy and homotopy equivalence
  ...quivalence relation that produces the same result for a much broader class of spaces: ...try to understand the actual mathematics behind these words, with the help of this juxtaposition:
  45 KB (7,738 words) - 15:18, 24 October 2015
• Integer-valued calculus
  <blockquote>''God made the integers, all else is the work of man.'' -- Leopold Kronecker </blockquote> ...''orientable''? Can one go on a trip and then come back as a mirror image of himself? Including the internal organs, like this:
  27 KB (3,824 words) - 19:07, 26 January 2019
• Cubical complexes
  ...Z}^N$. Suppose also that we have its decomposition ${\mathbb R}^N$, a list of all (closed) cubical cells in our grid. ...which the boundary operator is well-defined; i.e., $K$ includes all faces of the cells it contains.
  20 KB (3,319 words) - 14:18, 18 February 2016
• Contemporary Abstract Algebra by Gallian
  ''Contemporary Abstract Algebra'' by Joseph A. Gallian Used the book (twice) -- for [[Modern Algebra I -- Fall 2011]], see also [[Group theory: course]].
  5 KB (568 words) - 15:23, 16 November 2011
• Cochains on graphs
  ==The algebra of plumbing, continued== We introduce more algebra with the familiar metaphor:
  15 KB (2,523 words) - 18:08, 28 November 2015
• Calculus on graphs
  ==The algebra of plumbing, continued== We introduce more algebra with the familiar metaphor.
  16 KB (2,578 words) - 00:14, 18 February 2016
• Exterior derivative
  ==What may be the meaning of the derivative of a differential form?== There are a couple of questions to consider.
  12 KB (2,089 words) - 18:16, 22 August 2015
• Discrete parametric curves
  that comes from Pythagorean Theorem. diagonal of box $a \times b \times c$.
  32 KB (5,426 words) - 21:57, 5 August 2016
• Integration of differential forms of degree 2
  It is linear in either of the two arguments but not linear in $(a,b)$: ...d it is differentiable with respect to $a$ and linear with respect of each of the vectors $v$.
  15 KB (2,341 words) - 20:53, 13 March 2013
• Stability Analysis
  ...numerical scheme to discretize the equation, and we also have STABILITY. Of course, stability will turn out to depend on the parameter values delta-t a ...components of our separated solution; using this, and a clever trick from algebra, we can finally prove our stability condition given above!
  12 KB (2,051 words) - 03:51, 11 August 2012
• Calculus I -- Fall 2017
  ...ions of one variable, including the transcendental functions. (PR: MTH ACT of 27 or above, or MTH 130 and 122, or MTH 127 and 122, or MTH 132) This cours *Prerequisites: fluency with algebra, good understanding of functions.
  13 KB (2,075 words) - 13:35, 27 November 2017
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...In contrast to traditional goals of finding an accurate ''discretization'' of conventional multivariate calculus, discrete calculus establishes a separat ...complex]] in terms of $(k-1$-cells, you also know the exterior derivative of all discrete [[differential forms]] ([[co-chain]]s). So, you know calculus.
  11 KB (1,663 words) - 16:03, 26 November 2012
• Notation
  *$A^n:=\{(x_1,...,x_n):x_i\in A\} \quad$ the $n$th power of set $A$; ...X:A \hookrightarrow X \quad$ the inclusion function of subset $A\subset X$ of set $X$ into $X$;
  8 KB (1,519 words) - 16:30, 1 December 2015
• College Algebra -- Fall 2016
  ...ynomial, rational, exponential, and logarithmic functions. Graphs, systems of equations and inequalities, sequences. (PR: Math ACT 21 or above) *Prerequisites: solid algebra skills, some knowledge of Cartesian coordinates, familiarity with basic functions
  10 KB (1,078 words) - 19:07, 16 December 2016
• Links
  *[[Calculus as a part of topology]] *[[Multilinear algebra]]
  6 KB (998 words) - 12:40, 31 August 2015
• Calculus I -- Spring 2017
  ...ions of one variable, including the transcendental functions. (PR: MTH ACT of 27 or above, or MTH 130 and 122, or MTH 127 and 122, or MTH 132) This cours *Prerequisites: fluency with algebra, good understanding of functions
  12 KB (1,803 words) - 20:50, 1 May 2017
• Calculus I -- Fall 2016
  ...ions of one variable, including the transcendental functions. (PR: MTH ACT of 27 or above, or MTH 130 and 122, or MTH 127 and 122, or MTH 132) This cours *Prerequisites: fluency with algebra, good understanding of functions
  11 KB (1,671 words) - 23:11, 13 December 2016
• Fantasy math
  I would like to come up with an outline of what undergraduate mathematics curriculum ought to be. I have to ''ignore the reality'' of college education: prerequisites, "service courses", overlapping degree req
  8 KB (1,196 words) - 13:02, 24 August 2015
• Discrete differential forms
  [[Image:graph of function.png|center]] ...ts values at, say, 0, 1, 2, 4, 5... Suppose that the [[derivative]], $f'$, of the function is also sampled:
  10 KB (1,471 words) - 12:50, 12 August 2015
• Calculus by Stewart
  ...rd, there is nothing wrong with the book. The "Early Transcendentals" part of the title has always bugged me though... A preview of calculus.
  6 KB (794 words) - 16:29, 13 August 2017
• Fundamental group
  The ''fundamental group'' of $(X,x_0)$, $\pi _1(X,x_{0})$, is the [[group]] of loops based at $x_{0}$, up to a based [[homotopy]]. We define a product on where $<x>$ stands for the homotopy class of $x$ and $\sigma \cdot \tau$ means "travel along $\sigma$ and then $\tau$".
  10 KB (1,673 words) - 18:23, 2 December 2012
• Applied Calculus -- Spring 2015
  '''MTH 140 - Applied Calculus''' A brief survey of calculus including both differentiation and integration with applications. ...classes of functions, graph these functions, solve equations -- [[College Algebra -- Fall 2014]]
  9 KB (1,141 words) - 16:08, 26 April 2015

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