Intereting Posts

If $\alpha=\frac{2\pi}{7}$,prove that $\sin\alpha+\sin2\alpha+\sin4\alpha=\frac{\sqrt7}{2}$
The values of the derivative of the Riemann zeta function at negative odd integers
Mean value theorem for integrals: how does the sign matter?
Riemann Lebesgue Lemma Clarification
Find a Recurrence Relation
Infinite Series: Fibonacci/ $2^n$
Prove $\sin(x)< x$ when $x>0$ using LMVT
Applications of descriptive set theory to mathematical logic?
Relationship between Cyclotomic and Quadratic fields
Can this inequality proof be demystified?
What is the meaning of set-theoretic notation {}=0 and {{}}=1?
Solution to exponential congruence
Is it formally correct the acceptance-rejection question?
roots of a polynomial inside a circle
Limit of product with unbounded sequence $\lim_{n\to\infty} \sqrt{n}(\sqrt{n}-1)=0$

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.

- Show that every k-dimensional vector subspace V of $R^N$ is a manifold diffeomorphic to $R^k$.
- Tensors as mutlilinear maps
- On surjectivity of exponential map for Lie groups
- Find the geodesics on the cylinder $x^2+y^2=r^2$ of radius $r>0$ in $\mathbb{R}^3$.
- How should I think about what it means for a manifold to be orientable?
- Differential topology book
- Elementary proof of the fact that any orientable 3-manifold is parallelizable
- Flow of sum of non-commuting vector fields
- The number of geodesics of a complete Riemann manifold with non-positive sectional curvature
- Deriving generators for $H^1(T)$: what are $dx$ and $dy$?

Consider the vector space generated by the velocity vectors at $0$ of all curves passing through the origin: it is of dimension $3$. If the cone were an inmersed submanifold, its dimension should therefore be $3$ and then it would have to have a non-empty interior.

(This is the idea of a proof: you should actually prove every statement I made!)

Recall that a function $f:\mathbb R^2 \to \mathbb R^3$ is called an *immersion* if $f$ is differentiable and the derivative $Df$ is injective at every point of $\mathbb R^2$. There are then a few problems here, the first being that $z$ is not a function of $x$ and $y$ (it has two values, $-\sqrt{(x^2 + y^2)}$ and $\sqrt{(x^2 + y^2)}$). Setting this aside, you may take the “upper half” of the cone $$z = \sqrt{(x^2 + y^2)}$$

so that now $z$ is a function of $x$ and $y$, however this is still not an immersion since $Dz$ is not defined at the point $(0,0)$. One way to see this is by writing down the definition (using a limit) of $Dz$ and showing that the limit doesn’t exist. However it is easier to see this geometrically. You can make a paper model: cut out a disk with some paper, then cut out a wedge (“pizza slice”) from the disk. Then take the disk with the sliced removed and glue along the edges where you cut out the wedge. You have constructed the image of the function $z$, and you can see that it has a singular nonsmooth point, being the vertex of the cone.

- Is this “theorem” true in Optimization Theory?
- Proof for $A \cup B = B$ if and only if $A \subset B$
- How “Principia Mathematica” builds foundations
- Does $\pi$ have infinitely many prime prefixes?
- Basis of a vector space is a maximal linearly-independent set?
- How to evaluate the derivatives of matrix inverse?
- Normal subgroups of dihedral groups
- How to solve a cyclic quintic in radicals?
- combining conditional probabilities
- Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
- Show that $f^{n}(0)=0$ for infinitely many $n\ge 0$.
- Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?
- Divisibility of binomial coefficient by prime power – Kummer's theorem
- $\forall x,y>0, x^x+y^y \geq x^y + y^x$
- Is it always true that $(A_1 \cup A_2) \times (B_1 \cup B_2)=(A_1\times B_1) \cup (A_2 \times B_2)$