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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
• 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
• 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
• 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

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