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

#### swampwiz

They all seem to mean the same thing. I personally have been using locus.

#### WWGD

Gold Member
What kind of graph are you referring to? A collection of vertices, edges with an incidence relation or the graph of a function, etc?

#### jedishrfu

Mentor
Where are you getting these questions from? Are you taking a course in Abstract Algebra or self studying math in general?

#### Mark44

Mentor
They all seem to mean the same thing. I personally have been using locus.

#### swampwiz

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.

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#### swampwiz

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.

#### swampwiz

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.

#### fresh_42

Mentor
2018 Award
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.
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.

#### swampwiz

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.

#### fresh_42

Mentor
2018 Award
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 $f$.

#### swampwiz

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?

#### WWGD

Gold Member
A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.

#### swampwiz

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.

#### fresh_42

Mentor
2018 Award
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.

#### fresh_42

Mentor
2018 Award
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.

#### swampwiz

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.

#### WWGD

Gold Member
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.

#### fresh_42

Mentor
2018 Award
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.

#### Mark44

Mentor
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.

#### WWGD

Gold Member
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.

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

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