# Tagged Questions

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### trace map is continuous

Prove that $tr: M_n(k)\to k$ is continuous. I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good. ...
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### $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
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### A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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### Cont. Function smooth iff composition with submanifold inclusion is smooth

I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then ...
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### Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
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### finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
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### Diffeomorhism of manifold

This is one of the exam questions of the previous semester. I have studied these. But I didn't do this. Please show me how to solve this question. Thank you for help
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### Problem about tangent vector and the inclusion map of the unit circle.

It is so complecated for me. Please can you show how to solve. Thank you.
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### The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $∂/∂u|p,∂/∂v|p$ into $TpR_{3}.$ Find $(α_{i}, β_{i}, γ_{i})$

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
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### Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
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### Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
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### Is $S$ a regular submanifold of $\Bbb R^{3}$?

$$S=\{(x,y,z) \mid x^{2}+y^{2}=z^{2}\}$$ $g: \Bbb R^{3}\to \Bbb R$, $S=g^{-1}(0)$ Is $S$ a regular submanifold of $\Bbb R^{3}$? I'd be grateful for a clear and explicit explanation of why this is ...
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### Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
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### 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 ...
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