Tag Info

Hot answers tagged

7

This is Gauss' Linking Number Formula, for two space curves $\vec{A}, \vec{B}: S^1 \to \mathbb{R}^3$ $$ \textrm{link}(A,B) = \oint_A \oint_B \frac{\vec{A}-\vec{B}}{|\vec{A}-\vec{B}|^3} \cdot (d\vec{A} \times d\vec{B})$$ In our case, $\vec{A}(t) = (\cos t, \sin t, 0)$ and $\vec{B}(t) = ( 1+ \cos 2t, \frac{1}{2}\sin t, \sin 2t)$ . How to picture these two ...


7

$\infty$ and $-\infty$ are not real numbers. So when ever we use them in real analysis we need to define their use carefully, as he has done in the extract you quote. For instance the interval $(a,b)$ is defined as $\{ x \in \mathbb R \ : \ a < x < b \}$ whenever $a, b$ are both real numbers. This definition is in trouble if the symbols $a$ or $b$ ...


4

In short, the symbol has separate but analogous meanings in different contexts. While we often use it in place of a real number in notation, careful usage will not treat it as a real number---consider, for example, the notation $\lim_{x \to \infty} f(x)$. To expand orthogonally to Simon S' excellent answer (and perhaps besides the cursory paragraph above, ...


4

If the goal is to be readable and understandable by others: use the standard notations. So that people don't have to put too much effort in order to understand your equations. Attention $$\text{compact notation $\neq $ easy to read}$$ Sometimes it helps, sometimes it just makes things worst even more if the reader is not familiar with the notation used in ...


3

Here is 'physical' derivation for Gauss's linking number formula. In contrast with David H's remark above, I will talk in terms of magnetic circulation rather than the magnetic force between two current-carrying wires. Suppose I have two closed curves, with paths $C_1,C_2$ respectively. We can generate a magnetic field by running a uniform current $I$ ...


3

You could put it this way: If we write $$\begin{align}\lim_{x\to a}f_1(x)&=b\\ \lim_{x\to c}f_2(x)&=\infty\\ \lim_{x\to \infty}f_3(x)&=d\\ \end{align}$$ then there is a qualitative difference between these three lines. That is, the second is not just the same as the first with $c$ in place of $a$ and $f_2$ in place of $f_1$ and $\infty$ in ...


2

It is a subscript, indicating that $q_1$ is not necessarily the same as $q$. -- hardmath The author has established that $p^2$ is divisible by $4$. This makes it possible to write $p^2=4q_1$ where $q_1$ is some integer. One could use another letter, say, "$p^2=4r$ where $r$ is an integer". But if this process continues, one runs out of letters very ...


2

For any set $S$, we have the following two notations: $$|S|=\text{the cardinality of }S\qquad\qquad S^2=S\times S=\{(a,b):a,b\in S\}$$ Thus, $|X^2|$ is the cardinality of the set of ordered pairs of elements of $X$. which, incidentally, is equal to $|X|^2$. For example, if $X$ were a set with $2$ elements, then $|X^2|=4$.


2

I'd say: $$r=z^{\frac{1}{n}}e^{\frac{2i\pi k}{n}}$$ It is a multivalued function with $k=0,\dots,n-1$


2

The implication is a consequence of Ricci identity. The notations $\nabla_k$ is standard: Let, in general, that $A$ is a $(p, q)$-tensor, one can define a $(p, q+1)$ tensor $\nabla A$. In your specific example, there are two tensors: The function, $\omega$, which is a $(0,0)$ tensor, and It's exterior derivative $d\omega$, which is a one form (that is, a ...


1

It's a Neumann boundary condition. Here $n$ stands for the outward-pointing unit normal vector on the boundary of your domain and $\nabla u \cdot n = \sum_{i=1}^2 \partial_{x_i} u n_i$ is the dot product of the gradient, $\nabla u$, and the vector $n$.


1

For 2, if the collection is finite, then you can consider its interval graph, which is a graph representation of the intersections among the intervals in the collection. Now I quote Wikipedia: The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs.


1

No, $\partial x$ cannot be understood as a product. If it could be, then you would get $$\frac{\partial y}{\partial x} = \frac{y}{x}$$ which obviously is not true in general. The expression $\frac{\partial}{\partial x}$ is also known as derivation operator. This can be understood as follows: Given a function of several variables, say $f(x,y)$, partial ...


1

As Jonathan Hebert noted, $\Bbb R / 2\pi$ is probably the set of real numbers modulo $2\pi$. Are you familiar with $\Bbb Z_{n}$ from Abstract Algebra? It's the set of integers modulo the natural number $n$, and the concept here is the same. Basically, you look at $\Bbb R$ and for each $x \in \Bbb R$, you start forming its equivalence class. You say $y ...


1

In the sense that Spivak discusses (i.e. the way it is generally used in calculus), "$\infty$" is really a sort of abbreviation to allow you to write certain limits and sets in a way consistent with other sets. Here are a couple of examples: The interval $(a,\infty)$ is the set of real numbers that are larger than $a$. This is by analogy with the interval ...



Only top voted, non community-wiki answers of a minimum length are eligible