What's the difference between graph, locus & manifold?

In summary, the conversation touches on various mathematical concepts, including graphs, loci, manifolds, rings, and fields. The distinction between these concepts is explained, with a focus on their properties and relationship to each other. The conversation also delves into Galois theory and the idea of root values as allowable operations in a group.
  • #1
swampwiz
571
83
They all seem to mean the same thing. I personally have been using locus.
 
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  • #2
What kind of graph are you referring to? A collection of vertices, edges with an incidence relation or the graph of a function, etc?
 
  • #3
Where are you getting these questions from? Are you taking a course in Abstract Algebra or self studying math in general?
 
  • #4
swampwiz said:
They all seem to mean the same thing. I personally have been using locus.
Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.
 
  • #5
jedishrfu said:
Where are you getting these questions from? Are you taking a course in Abstract Algebra or self studying math in general?

I'm self-studying math in general. I have taken the standard mathematics that a physics/engineer takes - standard calculus, multi-dimensional & vector calculus, series, linear algebra, ordinary differential equations, partial differential equations, LaPlace & Fourier transforms. I also took a course in calculus of variations, but I feel like I didn't learn anything in that other than the Euler-Lagrange formula; I plan to start asking questions on that on this forum later on. :oldbiggrin:
 
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  • #6
WWGD said:
What kind of graph are you referring to? A collection of vertices, edges with an incidence relation or the graph of a function, etc?

What I mean is the collection of points that correspond to some continuous function (whether it is known or unknown) in some domain space.

I think I know what you are talking about in terms of a discrete graph such as would be encountered in computer science; this is not what I am referring to here.
 
  • #7
Mark44 said:
Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.

I did research it, but it just confused me since they seem to mean almost the same thing. And reading through "Galois theory" by Stewart, the discussion seems to go back & forth between fields & rings, which just totally confused me. I have not had any coursework in abstract algebra.
 
  • #8
swampwiz said:
What I mean is the collection of points that correspond to some continuous function (whether it is known or unknown) in some domain space.

I think I know what you are talking about in terms of a discrete graph such as would be encountered in computer science; this is not what I am referring to here.
The difficulties with your questions are that they are not precise and confuse different things, so they would need an entire lecture to explain them, and that Wikipedia should already have solved much of them.

A graph ##G## of a function ##f\, : \,x \longmapsto f(x)## is simply a set ##G=\{\,(x,f(x)) \,|\, x \in \operatorname{dom}(f)\,\}##.

A locus is no mathematical term. Better would be location, but in the end it is just a point which is not further qualified, and its use is always dependent on context.

A manifold is a complex mathematical structure with various properties. A graph can be a manifold, but a manifold doesn't have to be defined by a single function. Manifolds range from spheres, tori or certain groups, but can also be e.g. a cube. Usually it is not a cube because some kind of differentiability is assumed in most cases, which automatically requires a differentiable structure, i.e. many more properties. Something a graph doesn't have to possess.
swampwiz said:
I did research it, but it just confused me since they seem to mean almost the same thing. And reading through "Galois theory" by Stewart, the discussion seems to go back & forth between fields & rings, which just totally confused me. I have not had any coursework in abstract algebra.
Rings: ##\mathbb{Z}\, , \,\mathbb{Z}_n\, , \,\mathbb{F}[x_1,\ldots,x_n]\, , \,\mathbb{M}_n(\mathbb{F})## and a lot more.
Fields: ##\mathbb{Q}\, , \,\mathbb{R}\, , \,\mathbb{C}\, , \,\mathbb{Z}_p\, , \,\mathbb{F}(x_1,\ldots,x_n)## and many more.

Fields allow divisions, rings usually do not. Some rings (integral domains like ##\mathbb{Z}## or ##\mathbb{F}[x]##) allow an extension into fields (##\mathbb{Q}##, resp. ##\mathbb{F}(x)##), some do not, e.g. ##\mathbb{Z}_6##. Galois theory is based on finite (automorphism) groups and field extensions (e.g. ##\mathbb{Q}\subsetneq \mathbb{Q}[\sqrt{3}] =\mathbb{Q}(\sqrt{3})##).

You see, as soon as I try to answer your question, as soon it provokes even more questions and a complete answer would result in a lecture. Something which we cannot provide.
 
  • #9
Wikipedia says:
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

This sounds like the proper definition. It sounds like a manifold is some abstract form of a locus, which is deeper than the knowledge I am seeking.
 
  • #10
swampwiz said:
This sounds like the proper definition. It sounds like a manifold is some abstract form of a locus, which is deeper than the knowledge I am seeking.
It sounds, but it isn't: "whose location satisfies or is determined by one or more specified conditions".
This is so vague, that it is impossible to work with. Geometric loci with emphasis on geometric build a geometric object as a line, o.k., but this cannot be used outside of this context. The main difference is, that we do not have coordinates in pure geometry!

Manifolds have points which make them, and these points as a whole do indeed have clearly defined and specified analytical properties with emphasis on analytical.

A graph is what I had defined above, so also precisely defined. And we do not have any restrictions on the function ##f##.
 
  • #11
fresh_42 said:
Rings: ##\mathbb{Z}\, , \,\mathbb{Z}_n\, , \,\mathbb{F}[x_1,\ldots,x_n]\, , \,\mathbb{M}_n(\mathbb{F})## and a lot more.
Fields: ##\mathbb{Q}\, , \,\mathbb{R}\, , \,\mathbb{C}\, , \,\mathbb{Z}_p\, , \,\mathbb{F}(x_1,\ldots,x_n)## and many more.

Fields allow divisions, rings usually do not. Some rings (integral domains like ##\mathbb{Z}## or ##\mathbb{F}[x]##) allow an extension into fields (##\mathbb{Q}##, resp. ##\mathbb{F}(x)##), some do not, e.g. ##\mathbb{Z}_6##. Galois theory is based on finite (automorphism) groups and field extensions (e.g. ##\mathbb{Q}\subsetneq \mathbb{Q}[\sqrt{3}] =\mathbb{Q}(\sqrt{3})##).

I've noticed that somehow the idea of allowing the value of a root of an integer to the set of allowable numeric values is key to understanding Galois Theory, as in once such a value is used in a quintic polynomial, it is impossible for the result to be rational numbers. (I'm going through all the little theorems & lemmas in the Stewart book very carefully.) And I can see how arithmetic operations with the arguments being the rational numbers result in rational numbers, and not irrational numbers. So a field basically means a set of numbers that can have arithmetic operations done on them such that the results stay in the field; is this what is known as "closed field"?

BTW, what is the difference between a theorem, lemma & corollary? Is it that a theorem requires a major proof, while a lemma or corollary only requires a minor proof that the reader could figure out on his own?
 
  • #12
A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.
 
  • #13
WWGD said:
A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.

So a manifold is like a locus that mathematicians do topological stuff to.
 
  • #14
WWGD said:
A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.
I don't think that is a good picture because it sounds like a chain complex, not a manifold, at least not a differentiable one. A cube is a manifold in its most abstract sense, yes, but what you described sounds as if only such objects were manifolds, but they are not what is normally associated with the term.
 
  • #15
swampwiz said:
So a manifold is like a locus that mathematicians do topological stuff to.
No. no, no. Firstly, the comparison is a very bad one, and secondly, read my posts: analytical, not topological. Although what @WWGD described was more a topological object rather than a manifold.
 
  • #16
fresh_42 said:
No. no, no. Firstly, the comparison is a very bad one, and secondly, read my posts: analytical, not topological. Although what @WWGD described was more a topological object rather than a manifold.

OK, a locus is a just the most basic of manifolds - a bunch of points with nothing special going on.
 
  • #17
swampwiz said:
OK, a locus is a just the most basic of manifolds - a bunch of points with nothing special going on.
No, they actually do have structure, otherwise they would just be a set.
 
  • #18
swampwiz said:
So a field basically means a set of numbers that can have arithmetic operations done on them such that the results stay in the field; is this what is known as "closed field"?
No. Closed alone makes no sense, i.e. needs a specification by an adverb or by context. You used closed as closed under arithmetic operations, but this is trivially given by definition of a field. In Galois theory we consider algebraically closed fields, which are fields in which any polynomial ##p(x)=a_n+a_{n-1}x+\ldots + a_1x^{n-1}+x^n## can be written as ##p(x)=(x-\alpha_1)\cdot \ldots \cdot (x-\alpha_n)##.
BTW, what is the difference between a theorem, lemma & corollary? Is it that a theorem requires a major proof, while a lemma or corollary only requires a minor proof that the reader could figure out on his own?
These are conventions. A theorem is a major statement, a Lemma something which helps to prove theorems but often have a meaning on its own. Some Lemmata are called as such for historic reasons. A corollary is Latin and means a "gift", i.e. it is something which drops out of a theorem for free, or with only few additional arguments.
 
  • #19
Mark44 said:
Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.
swampwiz said:
I did research it, but it just confused me since they seem to mean almost the same thing.
I provided definitions of rings and fields in the other thread. If your research had been more careful, you would have seen that these terms didn't mean the same thing.
 
  • #20
swampwiz said:
OK, a locus is a just the most basic of manifolds - a bunch of points with nothing special going on.
A locus is just a collection of points determined by some condition. This is way too general. There is a very quick diminishing return of doing things so casually. There is a good reason why precise technical definitions are used.
 
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  • #21
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  • #22
fresh_42 said:
No. Closed alone makes no sense, i.e. needs a specification by an adverb or by context. You used closed as closed under arithmetic operations, but this is trivially given by definition of a field. In Galois theory we consider algebraically closed fields, which are fields in which any polynomial ##p(x)=a_n+a_{n-1}x+\ldots + a_1x^{n-1}+x^n## can be written as ##p(x)=(x-\alpha_1)\cdot \ldots \cdot (x-\alpha_n)##.

OK, I think I understand this now. If I have an established field F and I add a new number that is a root G(1/n) as a field extension to F, because I can do any of the arithmetic operations, adding the new number G is equivalent to adding any number that would result from using that number G(1/n) as the argument for a polynomial having coefficients that are in F. This is so because the only extra numbers that are added are those that have terms that contain the original numbers in F and any power of G(1/n), or G(k/n), with k being in the range [0,n) since the power of a root repeats as per a DeMoivre root around a circle.
 
  • #23
fresh_42 said:

A locus can be more than 1 dimension. Perhaps it is that a surface manifold has a 2-dimensional locus of points for the surface and 1-dimensional locus for the edge of the surface? This sounds like the idea of the surface integral used in Maxwell's Equations.
 
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  • #24
swampwiz said:
A locus can be more than 1 dimension. Perhaps it is that a surface manifold has a 2-dimensional locus of points for the surface and 1-dimensional locus for the edge of the surface? This sounds like the idea of the surface integral used in Maxwell's Equations.
Sure. That was an illustration of the basic usage. I could as well have given you the correct definitions, but
  1. Wikipedia already has them.
  2. You don't read them anyway:
    Mark44 said:
    I provided definitions of rings and fields in the other thread. If your research had been more careful, you would have seen that these terms didn't mean the same thing.
swampwiz said:
OK, I think I understand this now. If I have an established field F and I add a new number that is a root G(1/n) as a field extension to F, because I can do any of the arithmetic operations, adding the new number G is equivalent to adding any number that would result from using that number G(1/n) as the argument for a polynomial having coefficients that are in F. This is so because the only extra numbers that are added are those that have terms that contain the original numbers in F and any power of G(1/n), or G(k/n), with k being in the range [0,n) since the power of a root repeats as per a DeMoivre root around a circle.
This is also half baked and needed precision. The notation ##G^{1/n}## should be avoided, it is misleading. Maybe you should first examine how we constructed the following tower:
$$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{Q}[\sqrt[5]{7},\sqrt{2}] \subseteq \mathbb{R} \subseteq \mathbb{C} \subseteq \mathbb{H}$$
and learn at each step what it is called: from half group to division ring.
 
  • #25
irresistible to try to answer such a vague question!

a locus is any set of points.

a graph (of a function) is a subset of a product space AxB which has at most one pair for each point of A.

the graph of a smooth function (defined on euclidean space and with values in a euclidean space) is an example of a (smooth) manifold.

In fact the previous example of a manifold is general in the sense that a (smooth sub-) manifold (of euclidean space) is everywhere locally just the graph of a smooth function.this has no overlap whatsoever with stewart's galois theory.
 

What is a graph?

A graph is a visual representation of data that shows the relationship between two or more variables. It consists of points or vertices connected by lines or edges.

What is a locus?

A locus is a set of points that satisfy a given condition or equation. It is often represented as a curve or surface in a graph.

What is a manifold?

A manifold is a mathematical concept that describes a space that locally resembles Euclidean space. It can be thought of as a higher-dimensional version of a curve or surface in a graph.

What are the main differences between a graph, locus, and manifold?

A graph is a visual representation of data, while a locus and manifold are mathematical concepts. A locus is a set of points that satisfy a given condition, while a manifold is a space that locally resembles Euclidean space.

How are graphs, loci, and manifolds used in science?

Graphs are commonly used to display and analyze data in various scientific fields. Loci and manifolds are used in mathematics and physics to describe and study complex systems and phenomena.

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