Intereting Posts

Proof that any simple connected graph has at least 2 non-cut vertices.
Sum of GCD(k,n)
Sum identity using Stirling numbers of the second kind
The Ext functor in the quiver representation
Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$
Prove or disprove: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$.
Continuous function and open set
How to evaluate limiting value of sums of a specific type
How to prove an identity (Trigonometry Angles–Pi/13)
“Pseudo-Cauchy” sequences: are they also Cauchy?
What is a homomorphism and what does “structure preserving” mean?
Problems on expected value
If $x^3 =x$ then $6x=0$ in a ring
Finding points on two linear lines which are a particular distance apart
Necessity/Advantage of LU Decomposition over Gaussian Elimination

Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an interval (not necessarily open or closed), and $f$ is smooth (i.e. infinitely differentiable)?

The reason I ask is that in my reading, I’ve come across two definitions for “line integral” (which I’m sure turn out to be equivalent):

The first is the standard definition given in most multivariable calculus and complex analysis classes, which relies on defining a “curve” as a (sufficiently nice) function (or image set) $z = \gamma(t)$. That is: $\int_\gamma f(z) dz = \int_a^b f(g(t))g'(t) dt$.

- Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$
- erf(a+ib) error function separate into real and imaginary part
- Geometric interpretation of complex path integral
- Image of the Riemann-sphere
- Demystifying modular forms
- The Schwarz Reflection Principle for a circle

The second is the more high-powered definition involving 1-manifolds and differential 1-forms.

So my question is not really about whether or not these two definitions are equivalent per se, but rather about how much generality is lost by looking at only the “special” 1-manifolds which admit representations as $z = \gamma(t)$.

(My own thoughts: (1) Do these 1-manifolds end up being the 1-manifolds with an atlas consisting of one chart? I’m thinking not, because the circle cannot be given an atlas with one chart, yet can still be parametrized by a single function. (2) Does the Implicit Function Theorem have any role to play?)

- Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$
- Analytic map with two fixed points on a simply connected domain is the identity
- A finite sum of reciprocals of complex numbers cannot be confined to a half-plane
- Show that if $|f(z)| \leq M |z|^n$ then $f$ is a polynomial max degree n
- Definite integral using the method of residues
- Rational function with absolute value $1$ on unit circle
- There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$
- Proving an entire, complex function with below bounded modulus is constant.
- inequality involving complex exponential
- Rational function which is bijective on unit disk

I believe this question has been answered:

several users commented on the fact that “every connected 1-manifold (with or without boundary) (not necessarily compact) is diffeomorphic to one of: $(0,1)$, $[0,1)$, $[0,1]$, and $\mathbb{S}^1$”, and Ryan Budney noted that “The proof that 1-manifolds are all of this form isn’t particularly hard, either. The idea is to orient a component then take the unit speed vector field on that component and use it to parametrize the component by arc length. You have to argue that such a parametrization is onto. Once you have that, the domain of the parametrization has to be a 1-dimensional submanifold of $\mathbb R$ which puts the problem into the land of single-variable calculus/analysis.”

- Proof for a particular integration result.
- Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area
- Question about the dirac $\delta$-function
- Subadditivity inequality and power functions
- To Solve a linear PDE of first order
- What is the probability density function of pairwise distances of random points in a ball?
- Proof that every number has at least one prime factor
- Winding Numbers and Fixed Point Theorems
- Is integration by substitution a special case of Radon–Nikodym theorem?
- Generalisation of euclidean domains
- Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?
- A Pick Lemma like problem
- Finding the “triangular root” of a number.
- Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$
- How to solve polynomial equations in a field and/or in a ring?