For requesting, clarifying, and comparing definitions of mathematical terms.

learn more… | top users | synonyms

6
votes
0answers
82 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
5
votes
3answers
2k views

The Degree of Zero Polynomial.

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
379
votes
21answers
65k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
-4
votes
4answers
502 views

Why do they call it base 10?

Now, I know intuitively why it's called base 10: because there's 10 numbers. But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at ...
2
votes
1answer
45 views

Is a hypersurface really defined by an arbitrary polynomial?

In An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...
2
votes
1answer
46 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) ...
0
votes
0answers
17 views

What is characteristic time?

What is characteristic time? Where is it useful? From this answer by Joriki: The characteristic time is usually defined to be the time in which a quantity decreases by $1/e$. Why is ...
3
votes
1answer
29 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
1
vote
0answers
17 views

Continuously variable *space*

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
5
votes
4answers
467 views

Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
4
votes
3answers
8k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
0
votes
1answer
43 views

Is continuity at a point only defined for points in the domain?

I'm using Michael Spivak's Calculus, 3rd edition textbook. Without ado, I'll state the definition given for continuity at a point: DEFINITION$\;\;\;\;$The function $f$ is continuous at $a$ if: ...
-2
votes
0answers
35 views

Liapunov stable and asymptotically stable [on hold]

Define what it means for a periodic orbit to be Liapunov stable, and asymptotically stable respectively. Can someone give simple defintions of these please. I have the definitions for when a set $A$ ...
0
votes
0answers
27 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
1
vote
0answers
23 views

Rewording the definition of closure

In Munkres there was a statement: Given a topological space $(X, \tau)$ $x \in \overline A \iff \text{ for every open set } U \text{ containing } x, U \cap A \neq \varnothing$ Following from ...
1
vote
0answers
41 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
1
vote
3answers
57 views

changing the order of the logical symbols $(\forall \epsilon) (\exists\delta)$ by $(\exists \delta)(\forall\epsilon) $ in limit definition

Some time ago a professor told the class, which I was in, to analyze why this definition of limit is not good (or if it is a good definition to argument why): There exists a $\delta>0$ for all ...
0
votes
1answer
48 views

What is the meaning of (resp. closed) in set theory?

I'm sure this a spectacularly basic question but I can't seem to find the definition of this anywhere. Here's some context: If $U$ and $V$ are open (resp. closed) then $U\cup V$ is open (resp. ...
0
votes
0answers
18 views

General definition of the discriminant

There is a notion of the discriminant of polynomials quadratic forms finite separable extensions of Dedekind domains (e.g., algebraic number fields) I don't know much about 2, but I think that 1 ...
1
vote
0answers
52 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
2
votes
1answer
28 views

Definition of component for a digraph?

I could find this in Wikipedia Component: A connected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have ...
1
vote
1answer
31 views

Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ ...
0
votes
1answer
19 views

Definition of vertex-cut for digraph?

I am trying to understand vertex cut for digraph. I could find this for graphs Vertex cut is a vertex whose removal increases the number of components in a graph. (D67, Handbook of Graph Theory by ...
1
vote
0answers
45 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as ...
3
votes
1answer
450 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
1
vote
1answer
49 views

Coordinate charts vs. coordinates on manifolds

I just realised that I'm confused what coordinates really means in the context of manifolds. For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ ...
1
vote
2answers
33 views

Interpretation of definitions and logical implication in Calculus - e.g. monotonic strictly increasing function

I read definitions in Calculus books that often confuse me from a logical perspective. For example, the definition of a monotonic function, e.g. a strictly increasing function, is defined as follows. ...
1
vote
1answer
48 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something ...
2
votes
0answers
23 views

Showing that an intersection of indexed sets is a subset of every individual indexed set

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$. This seems obvious to me from ...
24
votes
5answers
2k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
1
vote
0answers
25 views

w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?

This question is inspired by my previous question. People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. ...
8
votes
2answers
441 views

What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
1
vote
1answer
59 views

What does maximal order mean in a group?

As written in Abstract Algebra by T. W. Judson: Lemma 13.4 : Let $G$ be a finite abelian $p-$group and suppose that $g ∈ G$ has maximal order. Then $G$ is isomorphic to $g × H$ for some subgroup ...
16
votes
2answers
797 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
0
votes
1answer
9 views

How to show X is in the co-finite topology

This might be kind of a silly question but I can't fully grasp why the set X on which the cofinite topology is defined would be contained in the Topology. I know that the closure of all open sets U ...
-8
votes
0answers
30 views

The diagram commutes… [duplicate]

The diagram commutes... but it stays right there! What does this mean? I thought the diagram would spin but it remains completely stationary!
5
votes
1answer
466 views

Non-vanishing differential form: what does it mean?

A $1$-form $\alpha$ over a smooth manifold is non vanishing if for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is a linear map $T_p M\to \mathbb R$ hence $\alpha_p(0)=0$. So confusion arises ...
38
votes
20answers
2k views

Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines ...
0
votes
1answer
45 views

What is the formal definition of a limit at infinity?

I keep coming across two different kinds of answers to this question. The first definition: We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied: for every number ...
0
votes
1answer
51 views

Is the Identity Type an Identity Function?

Definition 1. Given a set $S$, the identity function on $S$ is the function $id_S:S \to S$ that maps any element $x \in S$ to itself. Proposition. Given a type $S$, the identity type for $S$ is the ...
2
votes
8answers
2k views

Seeking elegant proof why 0 divided by 0 does not equal 1

Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ...
2
votes
1answer
34 views

What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
0
votes
1answer
37 views

Understanding Stokes' theorem and the fundamental theorem of calculus

I don't know a lot about differentials and boundaries, so it may be out of my grasp, but is there perhaps a simple way of understanding the Stokes' theorem for the FTC? $\displaystyle\int_a^b ...
2
votes
3answers
147 views

Precise definition of free group

I have seen the definition of a free group go like this: Let $S = \{s_i : i\in \mathbb{N} \}$ be a countable set. Let $S^{-1}$ be the set $\{s_i^{-1}: i\in \mathbb{N}\}$. Here one is to understand ...
1
vote
2answers
86 views

Why do authors make a point of $C^1$ functions being continuous?

I've just got a little question on why authors specify things the way they do. Is their some subtlety I'm missing or are they just being pedantic? I've encountered the function spaces $C^k[a,b]$ a ...
0
votes
0answers
31 views

Derivative of function of one variable with respect to function of two variables

I'm looking to find the derivative of a function of one variable with respect to a function of two variables: $$ \frac{df(x)}{dg(x,y)} $$ I'm not entirely sure whether this is possible in the first ...
0
votes
1answer
27 views

Taylor series for exponential function.

The Taylor series for $e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$. Then as $e^0 = 1$, if one evaluates the Taylor series at $x=0$ we find that $e^0 = \sum_{i=0}^\infty \frac{0^i}{i!} = \frac{0^0}{0!} + ...
1
vote
0answers
41 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
1
vote
1answer
23 views

Different versions of Markov's inequality

I have some doubts regarding some different versions of Markov's inequality. From Wikipedia's definition of Markov's inequality we have: If $X$ is a nonnegative random variable and $a>0$, then ...
1
vote
1answer
90 views

How does one “join” two graphs in graph theory?

I am asked to find the join of two graphs in graph theory. But I cannot find the exact definition! I know that in lattice theory, we join every vertex of a graph to every vertex of another graph to ...