5
votes
4answers
249 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
3
votes
1answer
50 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
1
vote
0answers
40 views

Compute $\int_0^{x_0}f^\prime(x)$ and $\int_{x_0}^{x_1}f^\prime(x)$

Suppose $f: S^1 \to S^1$, and $f(x_i) = y$, where $x_i$s are the preimage of a regular value $y$. Then how can I compute $\int_0^{x_0}f^\prime(x), \int_{x_0}^{x_1}f^\prime(x),$? I realize that ...
0
votes
1answer
40 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
2
votes
2answers
54 views

$d_pf$ is a linear invertible map.

If $f: S_1\to S_2$ is a diffeomorphism, then $$d_pf: T_p(S_1)\to T_q(S_2)$$ is an invertible linear map and $$(d_pf)^{-1}=d_qf^{-1}$$ for any $p\in S_1$ $q\in S_2$ and $f(p)=q$ I cannot prove this ...
0
votes
1answer
45 views

Smooth function with equibounded family of derivatives

By $\mathcal{C}^{\infty}(\mathbb{R})$ we denote the space of smooth functions $\mathbb{R}\rightarrow \mathbb{R}$. Also, by $\mathrm{supp}(f)$ we denote the closure $\mathrm{Cl}(f^{-1}(\mathbb{R}_{\ne ...
1
vote
2answers
62 views

$\int_{S^1} \beta = 0 \Rightarrow \beta$ is the differential of a function. - Is this proof legit?

Let $\beta$ be a smooth $1$-form on $S^1$, and $\int_{S^1} \beta = 0.$ Prove $\beta$ is the differential of a function. I don't really have a clue for this question.. I am trying to follow Anthony ...
2
votes
1answer
242 views

Change of Variable vs. Change of Coordinates.

Are they the same thing? So given an example, I could work out by change of coordinates, but how can I apply Change of Variable to replace this process? Change of Variable in $\mathbb{R^k}$. ...
3
votes
1answer
120 views

$d\phi = \sum \frac{\partial \phi}{\partial x_i}dx_i.$

Just a work out for a very tautological question that I am very uncertain about. If $\phi: X \to \mathbb{R}$ is a smooth function, $d\phi_x: T_x(X) \to \mathbb{R}$ is a linear map at each point ...
0
votes
1answer
42 views

$1$-forms of coordinate function $dx_i(z)(a_1, \dots, a_k) = a_i$.

I am reading Guillemin and Pollack's Differential Topology Page 163: The coordinate functions $x_1, \dots, x_k$ on $\mathbb{R}^k$ yield $1$-form $dx_1, \dots, dx_k$ on $\mathbb{R}^k$. Check $dx_1, ...
4
votes
1answer
146 views

Closed and exact.

I tried this question, but I have no idea if I got it correctly. On $\mathbb{R}^2$, let $\omega = (\sin^4 \pi x + \sin^2 \pi(x + y))dx - \cos^2 \pi(x + y)dy$. Let $\eta$ be the unique $1$-form on ...
5
votes
2answers
68 views

$T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, $\operatorname{rank}(dT)=m$, show $T$ maps open sets to open sets.

Suppose $T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, with $dT$ having rank $m$ at all points in an open set $D \subset \mathbb{R}^n$. What is a proof that $T$ maps $D$ into an open set in ...
0
votes
1answer
68 views

Proof that $\operatorname{rank}(dT)=1$ implies the image is a curve

I have a question about the proof that if the differential $dT$ of a transformation has rank 1 (2) at each point in a domain, then the image will be a curve (surface). Stated more precisely (in ...
1
vote
0answers
87 views

Change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value.

Pictorially examine the catastrophic change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value, where $$f (x, y, z) = x^2 + y^2 - z^2.$$ I don't have the slightest idea ...
1
vote
1answer
106 views

The inverse of homogenous function

Given homogenous function $p$ with order $m$, how can I show that $$p^{-1}(a) = (\frac{a}{b})^{\frac{1}{m}}p^{-1}(b)?$$ The original question is: Let $p$ be any homogeneous polynomial in ...
0
votes
1answer
36 views

Check that $df_x(v) = (v,v).$

Here is a proof that I am totally different from my classmates'. So I am requesting for expert reference here. Thank you. :-) Let $f: X \rightarrow X \times X$ be the mapping $f(x) = (x,x).$ Check ...
0
votes
1answer
43 views

Smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$

Can I find a smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$, I cannot take $\gamma(t)=(t,\lvert t\rvert)$ as ...
1
vote
1answer
106 views

Fubini Theorem for measure zero

I know Fubini Theorem in calculus, but the measure zero version does not make sense to me: $n=k+1$, and $V_c$ is the "vertical slice" {c}$\times R_l$. Let $A$ be a closed subset of $R^n$ such that $A ...
0
votes
1answer
33 views

If $f$ is a Morse function, then so is $f \circ \phi^{-1}$, where $\phi: U \rightarrow \mathbb{R}^k$ is the coordinate chart.

I am trying to show: if when $f^\prime = 0$, then $f^{\prime\prime} \neq 0 \Leftrightarrow (f \circ \phi^{-1})^\prime = 0$, $(f \circ \phi^{-1})^{\prime\prime} \neq 0$. But the problem is, because ...
0
votes
1answer
58 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
2
votes
2answers
114 views

Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
2
votes
1answer
157 views

Equivalent definition of Tangent Spaces

There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ...
0
votes
1answer
113 views

smooth function $\mu:\mathbb{R}\rightarrow\!\mathbb{R}$ with $\mu(0)>\varepsilon$, $\:\mu_{[2\varepsilon,\infty)}=0$, $\:-1<\mu'\leq 0$

How can I prove (preferrably without the use of any heavy theorems) the existence of a smooth function $\mu\!:\mathbb{R}\rightarrow\!\mathbb{R}$ with properties $\mu(0)\!>\!\varepsilon$, ...
4
votes
2answers
132 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...
1
vote
1answer
90 views

Is there any connection between partial derivative and matrices?

I can see in some texts and books that the authors use big letters in order to describe partial derivative of function in $\mathbb{R^n}$ similar to the way we write matrices in linear algebra, for ...
3
votes
1answer
288 views

Is the notion of density really needed to define integration on nonorientable manifolds?

I am trying to understand, in as simple terms as possible: How to define integration for non-orientable manifolds, and why it is impossible to do so using only differential forms. In particular, ...