= Calculus
{wiki}
Well summarized as "the branch of mathematics that deals with [limits]".
= Mathematical analysis
{parent=Calculus}
{wiki}
= Analytical
{synonym}
An fancy name for , with the "more advanced" connotation.
= Limit
{disambiguate=mathematics}
{parent=Calculus}
{wiki}
= Limit
{synonym}
The fundamental concept of !
The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the , making the notion completely precise.
= Convergent series
{parent=Limit (mathematics)}
{wiki}
= Convergence
{disambiguate=mathematics}
{synonym}
= Converges
{disambiguate=mathematics}
{synonym}
= Convergent
{disambiguate=mathematics}
{synonym}
= Continuous function
{parent=Limit (mathematics)}
{wiki}
= Continuity
{synonym}
= Continuous
{synonym}
= Continuous problems are simpler than discrete ones
{parent=Continuous function}
This is a general philosophy that , and likely others, observes over and over.
Basically, , or higher order conditions like seem to impose greater constraints on problems, which make them more solvable.
Some good examples of that:
* complex problems:
*
* simple problems:
* characterization of
= Discrete
{parent=Continuous function}
Something that is very not .
Notably studied in .
= Discretization
{parent=Discrete}
{wiki}
= Discretize
{synonym}
= Infinity
{parent=Limit (mathematics)}
{title2=$\infty$}
{wiki}
= Infinite
{synonym}
\Q[Chuck Norris counted to infinity. Twice.]
= Finite
{synonym}
There are a few related concepts that are called infinity in :
* that are greater than any number
* the of a that does not have a finite number of elements
* in some number systems, there is an explicit "element at infinity" that is not a , e.g.
= L'Hôpital's rule
{parent=Limit (mathematics)}
{title2=limit of a ratio}
{wiki}
= Derivative
{parent=Calculus}
{wiki}
= Chain rule
{parent=Derivative}
{wiki}
Here's an example of the chain rule. Suppose we want to calculate:
$$
\dv{e^2x}{x}
$$
So we have:
$$
f(x) = e^x \\
g(x) = 2x
$$
and so:
$$
f'(x) = e^x \\
g'(x) = 2
$$
Therefore the final result is:
$$
f'(g(x))g'(x) = e^{2x} 2 = 2 e ^{2x}
$$
= Multivariable chain rule
{parent=Chain rule}
= Differentiable function
{parent=Derivative}
{wiki}
= Differentiable
{synonym}
= Differentiability
{synonym}
= Smoothness
{parent=Differentiable function}
{wiki}
= Infinitely differentiable function
{parent=Differentiable function}
= $C^{\infty}$
{synonym}
{title2}
= Bump function
{parent=Infinitely differentiable function}
{wiki}
= Flat top bump function
{parent=Bump function}
https://math.stackexchange.com/questions/1786964/is-it-possible-to-construct-a-smooth-flat-top-bump-function
= Maxima and minima
{parent=Derivative}
{wiki}
Given a $f$:
* from some space. For beginners the but more generally should work in general
* to the
we want to find the points $x$ of the of $f$ where the value of $f$ is smaller (for minima, or larger for maxima) than all other points in some of $x$.
In the case of , this problem is treated under the theory of the .
= Lifegard problem
{parent=Maxima and minima}
https://pumphandle.consulting/2020/09/04/the-lifeguard-problem-solved/
= Derivative test
{parent=Maxima and minima}
{wiki}
= Saddle point
{parent=Maxima and minima}
{wiki}
= Newton dot notation
{c}
{parent=Derivative}
= Partial derivative
{parent=Derivative}
{wiki}
= Partial derivative notation
{parent=Partial derivative}
= Partial derivative symbol
{parent=Partial derivative notation}
{title2=$\partial$}
Nope, it is not a , notably it is not a lowercase . It is just some random made up symbol that looks like a . Which is of course derived from , which is why it is all so damn confusing.
I think the symbol is usually just read as "" as in "d f d x" for $\pdv{F(x, y, z)}{x}$.
= Partial label partial derivative notation
{parent=Partial derivative notation}
{title2=$\partial_x F$}
{title2=$\partial_y F$}
= Partial index partial derivative notation
{parent=Partial derivative notation}
{title2=$\partial_0 F$}
{title2=$\partial_1 F$}
This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with as shown at {full}:
$$
\partial_0 F(x, y, z) = \pdv{F(x, y, z)}{x} \\
\partial_1 F(x, y, z) = \pdv{F(x, y, z)}{y} \\
\partial_2 F(x, y, z) = \pdv{F(x, y, z)}{x} \\
$$
This notation is similar to , but it uses indices instead of labels such as $x$, $y$, etc.
= Total derivative
{parent=Derivative}
{wiki}
The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.
E.g. in 1D:
$$
Total derivative = D[f(x_0)](x) = f(x_0) + \pdv{f}{x}(x_0) \times x
$$
and in 2D:
$$
D[f(x_0, y_0)](x, y) = f(x_0, y_0) + \pdv{f}{x}(x_0, y_0) \times x + \pdv{f}{y}(x_0, y_0) \times y
$$
= Directional derivative
{c}
{parent=Derivative}
{wiki}
= Integral
{parent=Calculus}
{wiki}
= Area
{parent=Integral}
{wiki}
= Volume
{parent=Area}
{wiki}
<3D> .
= Riemann integral
{c}
{parent=Integral}
{wiki}
The easy and less generic . The harder one is the .
= Lebesgue integral
{c}
{parent=Integral}
{wiki=Lebesgue_integration}
"More complex and general" integral. Matches the for "simple functions", but also [works for some "funkier" functions that Riemann does not work for].
sometimes wonders how much someone can gain from learning this besides , since we can hand-wave a on almost anything that is of practical use. The beauty is good reason enough though.
= Lebesgue integral vs Riemann integral
{c}
{parent=Lebesgue integral}
Advantages over Riemann:
* .
* https://youtu.be/PGPZ0P1PJfw?t=710 you are able to switch the order of integrals and limits of function sequences on non-uniform convergence. TODO why do we care? This is linked to the of course, but concrete example?
\Video[https://youtube.com/watch?v=PGPZ0P1PJfw]
{title=Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018)}
{description=
https://youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.
One advantage of that is that the range is always one dimensional.
But the main advantage is that having infinitely many discontinuities does not matter.
Infinitely many discontinuities can make the Riemann partitioning diverge.
But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.
So then the question becomes more of "how to define the measure of a subset of the domain".
Which is why we then fall into !
}
= Real world applications of the Lebesgue integral
{parent=Lebesgue integral vs Riemann integral}
In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)
Its value is much more indirect and subtle, as in "it serves as a solid basis of " due to the definition of .
Bibliography:
* https://math.stackexchange.com/questions/53121/how-do-people-apply-the-lebesgue-integration-theory
* https://www.quora.com/What-are-some-real-life-applications-of-Lebesgue-Integration
= Lebesgue measurable
{c}
{parent=Lebesgue integral}
= Lebesgue integral of $\LP$ is complete but Riemann isn't
{c}
{parent=Lebesgue integral}
$\LP$ is:
* [complete] under the Lebesgue integral, this result is may be called the
* not complete under the : https://math.stackexchange.com/questions/397369/space-of-riemann-integrable-functions-not-complete
And then this is why basically lives in : not being complete makes no sense physically, it would mean that you can get closer and closer to states that don't exist!
TODO intuition
= Riesz-Fischer theorem
{c}
{parent=Lebesgue integral of LP is complete but Riemann isn't}
{wiki=Riesz–Fischer_theorem}
A measurable function defined on a closed interval is square integrable (and therefore in ) if and only if converges in norm the function:
$$
\lim_{N \to \infty} \left \Vert S_N f - f \right \|_2 = 0
$$
= $\LP$ is complete
{parent=Riesz-Fischer theorem}
TODO
= Fourier basis is complete for $\LTwo$
{id=fourier-basis-is-complete-for-l2}
{c}
{parent=Riesz-Fischer theorem}
https://math.stackexchange.com/questions/316235/proving-that-the-fourier-basis-is-complete-for-cr-2-pi-c-with-l2-norm
is a norm version of it, and is stronger pointwise almost everywhere version.
Note that the is weaker because the pointwise limit could not exist just according to it: .
= $L^p$ norm sequence convergence does not imply pointwise convergence
{id=lp-norm-sequence-convergence-does-not-imply-pointwise-convergence}
{parent=fourier basis is complete for l2}
https://math.stackexchange.com/questions/138043/does-convergence-in-lp-imply-convergence-almost-everywhere
There are explicit examples of this. We can have ever thinner disturbances to convergence that keep getting less and less area, but never cease to move around.
If it does converge pointwise to something, then it must match of course.
= Carleson's theorem
{c}
{parent=fourier basis is complete for l2}
{wiki}
The of an function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.
The theorem also seems to hold (maybe trivially given the transform result) for the (TODO if trivially, why trivially).
Only proved in 1966, and known to be a hard result without any known simple proof.
This theorem of course implies that , as it explicitly constructs a decomposition into the Fourier basis for every single function.
TODO vs . Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?
One of the many .
= Lp space
{parent=Lebesgue integral of LP is complete but Riemann isn't}
{wiki}
= $\LP$
{synonym}
{title2}
Integrable functions to the power $p$, usually and in this text assumed under the because:
= $L^1$
{id=l1-space}
{parent=Lp space}
= $\LTwo$
{id=l2}
{parent=Lp space}
<\LP> for $p == 2$.
$\LTwo$ is by far the most important of $\LP$ because it is [quantum mechanics states] live, because the total probability of being in any state has to be 1!
has some crucially important properties that other $\LP$ don't (TODO confirm and make those more precise):
* it is the only $\LP$ that is because it is the only one where an inner product compatible with the metric can be defined:
* https://math.stackexchange.com/questions/2005632/l2-is-the-only-hilbert-space-parallelogram-law-and-particular-ft-gt
* https://www.quora.com/Why-is-L2-a-Hilbert-space-but-not-Lp-or-higher-where-p-2
* , which is great for solving
= Plancherel theorem
{c}
{parent=l2}
Some sources say that this is just the part that says that the of a function is the same as the norm of its .
Others say that this theorem actually says that the is .
The comment at https://math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the statement is an easy consequence from the one, thus the confusion.
TODO does it require it to be in as well? https://en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but https://courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.
= The Fourier transform is a bijection in $L^2$
{parent=Plancherel theorem}
As mentioned at {full}, some people call this part of , while others say it is just a corollary.
This is an important fact in , since it is because of this that it makes sense to talk about as two dual representations of the that contain the exact same amount of information.
= Every Riemann integrable function is Lebesgue integrable
{parent=Plancherel theorem}
But only for the proper Riemann integral: https://math.stackexchange.com/questions/2293902/functions-that-are-riemann-integrable-but-not-lebesgue-integrable
= Measure theory
{parent=Calculus}
{wiki=Measure_(mathematics)}
Main motivation: .
The Bright Side Of Mathematics 2019 playlist: https://www.youtube.com/watch?v=xZ69KEg7ccU&list=PLBh2i93oe2qvMVqAzsX1Kuv6-4fjazZ8j
The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.
= Fourier series
{c}
{parent=Calculus}
{wiki}
Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.
's original motivation, and a key application, is .
Can only be used to approximate for periodic functions (obviously from its definition!). The however overcomes that restriction:
* https://math.stackexchange.com/questions/1115240/can-a-non-periodic-function-have-a-fourier-series
* https://math.stackexchange.com/questions/1378633/every-function-can-be-represented-as-a-fourier-series
The Fourier series behaves really nicely in , where it always exists and converges pointwise to the function: .
\Video[https://www.youtube.com/watch?v=r6sGWTCMz2k]
{title=But what is a ? by <3Blue1Brown> (2019)}
{description=Amazing 2D visualization of the decomposition of complex functions.}
= Applications of the Fourier series
{parent=Fourier series}
= Solving partial differential equations with the Fourier series
{parent=Applications of the Fourier series}
See: https://math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from .
of certain equations like the and are solved immediately by calculating the of initial conditions!
Other basis besides the Fourier series show up for other equations, e.g.:
*
*
= Discrete Fourier transform
{parent=Fourier series}
{title2=DFT}
{wiki}
Input: a sequence of $N$ $x_k$.
Output: another sequence of $N$ $X_k$ such that:
$$
x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{i 2 \pi \frac{k n}{N}}
$$
Intuitively, this means that we are braking up the complex signal into $N$ frequencies:
* $X_0$: is kind of magic and ends up being a constant added to the signal because $e^{i 2 \pi \frac{k n}{N}} = e^{0} = 1$
* $X_1$: that completes one cycle over the signal. The larger the $N$, the larger the resolution of that . But it completes one cycle regardless.
* $X_2$: that completes two cycles over the signal
* ...
* $X_{N-1}$: that completes $N-1$ cycles over the signal
and is the amplitude of each sine.
We use in our definitions because it just makes every formula simpler.
Motivation: similar to the :
* compression: a would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
* noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better
In particular, the is used in after a . historically likely grew more and more over analog processing as digital [processors] got faster and faster as it gives more flexibility in algorithm design.
Sample software implementations:
* , notably see the example: {file}
\Image[https://upload.wikimedia.org/wikipedia/commons/thumb/3/31/DFT_2sin%28t%29_%2B_cos%284t%29_25_points.svg/583px-DFT_2sin%28t%29_%2B_cos%284t%29_25_points.svg.png]
= Discrete Fourier transform of a real signal
{parent=Discrete Fourier transform}
See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal
The transform still has complex numbers.
Summary:
* $X_0$ is real
* $X_1 = \conj{X_{N-1}}$
* $X_2 = \conj{X_{N-2}}$
* $X_k = \conj{X_{N-k}}$
Therefore, we only need about half of $X_k$ to represent the signal, as the other half can be derived by conjugation.
"Representation in terms of sines and cosines" from https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of $X_k$.
for example has "Real FFTs" for this: https://numpy.org/doc/1.24/reference/routines.fft.html#real-ffts
= Normalized DFT
{parent=Discrete Fourier transform}
There are actually two possible definitions for the DFT:
* 1/N, given as "the default" in many sources:
$$
x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{i 2 \pi \frac{k n}{N}}
$$
* $1/\sqrt{N}$, known as the "normalized DFT" by some sources: https://www.dsprelated.com/freebooks/mdft/Normalized_DFT.html[], definition which we adopt:
$$
x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{i 2 \pi \frac{k n}{N}}
$$
The $1/\sqrt{N}$ is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.
* https://math.stackexchange.com/questions/3285758/scaling-magnitude-of-the-dft
* https://dsp.stackexchange.com/questions/63001/why-should-i-scale-the-fft-using-1-n
* https://www.dsprelated.com/freebooks/mdft/Normalized_DFT.html
= Fast Fourier transform
{parent=Discrete Fourier transform}
{wiki}
An efficient to calculate the .
= Fourier transform
{c}
{parent=Fourier series}
{wiki}
Continuous version of the .
Can be used to represent functions that are not periodic: https://math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the is only for periodic functions.
Of course, every function defined on a finite line segment (i.e. a ).
Therefore, the can be seen as a generalization of the that can also decompose functions defined on the entire .
As a more concrete example, just like the is how you solve the on a line segment with as shown at: {full}, the is what you need to solve the problem when the is the entire .
= Multidimensional Fourier transform
{parent=Fourier transform}
Lecture notes:
* http://www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf Lecture 2: 2D Fourier transforms and applications by A. Zisserman (2014)
\Video[https://www.youtube.com/watch?v=v743U7gvLq0]
{title=How the 2D FFT works by Mike X Cohen (2017)}
{description=Animations showing how the 2D Fourier transform looks like for simple inpuf functions.}
= Fourier inversion theorem
{parent=Fourier transform}
{wiki}
A set of theorems that prove under different conditions that the has an inverse for a given space, examples:
* for
= Laplace transform
{c}
{parent=Fourier transform}
\Video[https://www.youtube.com/watch?v=7UvtU75NXTg]
{title=The Laplace Transform: A Generalized Fourier Transform by Steve Brunton (2020)}
{description=Explains how the Laplace transform works for functions that do not go to zero on infinity, which is a requirement for the . No applications in that video yet unfortunately.}
= History of the Fourier series
{parent=Fourier series}
First published by Fourier in 1807 to solve the .
= Topology
{parent=Calculus}
{wiki}
= Topological
{synonym}
Topology is the plumbing of .
The key concept of topology is a .
Just by havin the notion of neighbourhood, concepts such as and can be defined without the need to specify a precise numerical value to the distance between two points with a .
As an example. consider the , which is also naturally a . That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that , and that .
= Covering space
{parent=Topology}
{wiki}
Basically it is a larger space such that there exists a from the large space onto the smaller space, while still being compatible with the of the small space.
We can characterize the cover by how injective the function is. E.g. if two elements of the large space map to each element of the small space, then we have a and so on.
= Double cover
{parent=Covering space}
= Neighbourhood
{disambiguate=mathematics}
{parent=Topology}
{wiki}
The key concept of .
= Topological space
{parent=Topology}
{wiki}
= Manifold
{parent=Topology}
{wiki}
We map each point and a small enough of it to <\R^n>, so we can talk about the manifold points in terms of coordinates.
Does not require any further structure besides a consistent map. Notably, does not require nor an addition operation to make a .
Manifolds are [cool]. Especially which we can do on.
A notable example of a manifold is the space of of a . For example, in a problem such as the , some of those generalized coordinates could be angles, which wrap around and thus are not .
= Atlas
{disambiguate=topology}
{parent=Manifold}
{wiki}
Collection of .
The key element in the definition of a .
= Coordinate chart
{parent=Atlas (topology)}
= Covariant derivative
{parent=Manifold}
{wiki}
A generalized definition of that works on .
TODO: how does it maintain a single value even across different ?
= Differentiable manifold
{parent=Manifold}
{wiki}
TODO find a concrete numerical example of doing on a differentiable manifold and visualizing it. Likely start with a boring circle. That would be sweet...
= Tangent space
{parent=Manifold}
{wiki}
TODO what's the point of it.
Bibliography:
* https://www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds \#6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
* https://www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the of any function along this direction".
One thing to remember is that of course, the most convenient way to define a function $f$ and to specify a direction, is by using one of the .
We can then just switch between charts by change of basis.
* http://jakobschwichtenberg.com/lie-algebra-able-describe-group/ by
* https://math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on
= Tangent vector to a manifold
{parent=Tangent space}
A member of a .
= One-form
{parent=Manifold}
{wiki}
https://www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a but for a : it measures how much that vector [derives] along a given direction.
= Metric
{disambiguate=mathematics}
{parent=Topology}
{title2=$d(x, y)$}
{wiki}
= Distance
{synonym}
= Metric
{synonym}
A metric is a function that give the distance, i.e. a , between any two elements of a space.
A metric may be induced from a as shown at: {full}.
Because a [norm can be induced by an inner product], and the given by the , in simple cases metrics can also be represented by a .
= Metric space
{parent=Metric (mathematics)}
{wiki}
Canonical example: .
= Metric space vs normed vector space vs inner product space
{parent=Metric space}
TODO examples:
* that is not a
* vs : a norm gives size of one element. A is the distance between two elements. Given a norm in a space with subtraction, we can obtain a distance function: the .
\Image[https://upload.wikimedia.org/wikipedia/commons/7/74/Mathematical_Spaces.png]
{title=Hierarchy of topological, metric, normed and inner product spaces}
= Complete metric space
{parent=Metric space}
{wiki}
In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.
One notable example where completeness matters: .
= Normed vector space
{parent=Metric space}
{wiki}
= Inner product space
{parent=Normed vector space}
{wiki}
Subcase of a , therefore also necessarily a .
= Inner product
{parent=Inner product space}
{wiki}
Appears to be analogous to the , but also defined for .
= Norm
{disambiguate=mathematics}
{parent=Metric space}
{title2=$|x|$}
= Norm
{synonym}
Vs :
* a norm is the size of one element. A is the distance between two elements.
* a norm is only defined on a . A could be defined on something that is not a vector space. Most basic examples however are also .
= Norm induced by an inner product
{parent=Norm (mathematics)}
{wiki}
= Norm induced by the inner product
{synonym}
An $x \cdot y$ induces a with:
$$
|x| = \sqrt{}
$$
= Metric induced by a norm
{parent=Norm (mathematics)}
In a , a may be induced from a norm by using :
$$
d(x, y) = |x - y|
$$
= Pseudometric space
{parent=Metric space}
{wiki}
but where the distance between two distinct points can be zero.
Notable example: {child}.
= Compact space
{parent=Topology}
{wiki}
= Compact
{synonym}
= Dense set
{parent=Topology}
{wiki}
= Connected space
{parent=Topology}
{wiki}
= Disconnected space
{synonym}
= Connected component
{parent=Connected space}
{wiki}
When a is made up of several smaller , then each smaller component is called a "connected component" of the larger space.
See for example the
= Simply connected space
{parent=Connected space}
{wiki}
= Simply connected
{synonym}
= Loop
{disambiguate=topology}
{parent=Simply connected space}
= Homotopy
{parent=Topology}
{wiki}
= Homotopic
{synonym}
= Generalized Poincaré conjecture
{parent=Homotopy}
There are two cases:
* (topological) manifolds
* differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
* for topological manifolds: this is a generalization of the .
Original problem posed, $n = 3$ for topological manifolds.
.
Last to be proven, only the 4-differential manifold case missing as of 2013.
Even the truth for all $n > 4$ was proven in the 60's!
Why is low dimension harder than high dimension?? Surprise!
AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
For dimension two, we know there are infinitely many:
* for differential manifolds:
Not true in general. First counter example is $n = 7$. Surprise: what is special about the number 7!?
Counter examples are called .
Totally unpredictable count table:
| Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Smooth types | 1 | 1 | 1 | ? | 1 | 1 | 28 | 2 | 8 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 | 523264 | 24 |
$n = 4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
= Exotic sphere
{parent=Generalized Poincaré conjecture}
{wiki}
= Poincaré conjecture
{c}
{parent=Generalized Poincaré conjecture}
{wiki}
= Classification of closed surfaces
{parent=Generalized Poincaré conjecture}
* https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces
* http://www.proofwiki.org/wiki/Classification_of_Compact_Two-Manifolds
So simple!! You can either:
* cut two holes and glue a handle. This is easy to visualize as it can be embedded in <\R^3>: you just get a , then a double torus, and so on
* cut a single hole and glue a in it. Keep in mind that this is possible because the has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
* 1:
* 2:
A handle cancels out a , so adding one of each does not lead to a new object.
You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!
Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.
= Torus
{c}
{parent=Classification of closed surfaces}
{wiki}
= Möbius strip
{c}
{parent=Classification of closed surfaces}
{wiki}
= Klein bottle
{c}
{parent=Classification of closed surfaces}
{wiki}
with two stuck into it as per the .
= Real coordinate space
{c}
{parent=Topology}
{wiki}
= $\R^n$
{synonym}
{title2}
= Real line
{parent=Real coordinate space}
{wiki}
= $\R^1$
{synonym}
{title2}
= 1D
{synonym}
= Real plane
{parent=Real coordinate space}
= $\R^2$
{synonym}
{title2}
= 2D
{synonym}
= Real coordinate space of dimension three
{c}
{parent=Real coordinate space}
= $\R^3$
{synonym}
{title2}
= 3D
{synonym}
= Real coordinate space of dimension four
{c}
{parent=Real coordinate space}
= $\R^4$
{synonym}
{title2}
= Four-dimensional space
{synonym}
= Four-dimensional
{synonym}
= 4D
{synonym}
{title2}
Important 4D spaces:
* <3-sphere>
= Visualizing 4D
{parent=Real coordinate space of dimension four}
Simulate it. Just simulate it.
\Video[http://youtube.com/watch?v=0t4aKJuKP0Q]
{title=4D Toys: a box of four-dimensional toys by Miegakure (2017)}
= Dimension
{parent=Real coordinate space}
{wiki}
= Infinite dimensional
{parent=Dimension}
= Infinite dimensions
{synonym}
https://math.stackexchange.com/questions/466707/what-are-some-examples-of-infinite-dimensional-vector-spaces
= Finite dimensional
{parent=Infinite dimensional}
= Finite dimension
{synonym}
= Complex coordinate space
{parent=Real coordinate space}
{wiki}
= $\C^n$
{title2}
{synonym}
= Complex coordinate space of dimension 2
{parent=Complex coordinate space}
= $\C^2$
{synonym}
{title2}
= Complex dot product
{parent=Complex coordinate space}
This section is about the definition of the over , which extends the definition of the over .
Some motivation is discussed at: https://math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169
The complex dot product is defined as:
$$
\sum a_i \overline{b_i}
$$
E.g. in $\C^1$:
$$
(a + bi) \cdot (c + di) = (a + bi) (\overline{c + di}) = (a + bi) (c - di) = (ac + bd) + (bc - ad)i
$$
We can see therefore that this is a