# Tagged Questions

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### How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
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### How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
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### Equation of tangent plane to a surface at certain point

So I am solving for an equation of a tangent plane to surface $z=x + \ln(2x+y)$ at the point $(-1,3,-1)$. I know I need to take partial derivative of that equation and plug in the point of the ...
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### how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
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### Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
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### The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
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### How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
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### Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...
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### Volume of “deformed torus”

I'm trying to find explicit form of volume of "deformed torus": Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : ...
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### How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
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### Derivative: a special tangent

I've learned in Euclidean Geometry that the tangent is a line which pass through only a point. For example, if someone ask me to find the tangent at this point $A$, I can easily say that the tangent ...
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### Why is the derivative the tangent vector?

I'm trying to understand, at least intuitively why the derivative of a function at a point is the tangent vector at this point. If we see the functions of this form $f:\mathbb R\to \mathbb R$ we see ...
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### Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
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### Find out the rate of change of rise of water at that moment.

The height and radius of a circular cone are $6\,\mathrm m$ and $8\,\mathrm m$ respectively. Water is pouring at a constant rate of $4\pi\,\mathrm{cm}^3/\mathrm s$ from another jar. Find out the ...
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### Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
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### Determine the number of revolutions the normal to a curve makes as it moves along a curve in three dimensions?

Given a point on a 3D curve, how many full revolutions does the normal to the curve at the point make as the point moves over the curve? Assume the point stops when it reaches the place where it ...
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### Find the sum of $\sum (n^2+n)x^n$ using integrals

I'm having a difficult to find $\sum_{n=1}^\infty (n^2+n)x^n$. the solution is $\frac{2x}{(1-x)^3}$. This is my solution: $$1. \space\space\space\space S(x) = \sum_{n=1}^\infty (n^2 +n)x^n =$$ ...
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### Calculation of $σ_u σ_u$ and $σ_u σ_v$

Accourding to the info which I posted, how can I calculate $σ_u σ_u=\vert\vert σ_u\vert\vert^2$ and $σ_u σ_v$ I am stuck with there. Please show me. Thanks.
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### Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
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### Show the curve $\alpha$ is differentiable and regular

Consider the map: $$\alpha(t) = \left\{ \begin{array}{ll} (t,0,e^{-1/t^2}) & t>0 \\ (t,e^{-1/t^2},0) & t<0 \\ (0,0,0) & t=0 \end{array} \right.$$ a. Prove that ...
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### Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
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### Change of Coordinates (Local Diffeo.) and its Effect on Geometry.

Everyone: Consider the (local) change of variables from Cartesian coordinates $(x,y,z)$ to Cylindrical coordinates $( r,\theta,z)$ given by f. Does this map preserve the local geometry; does it ...
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### Constant rank map means image locally a graph

Suppose that $f:U \to \mathbb{R}^n$, where $U$ is an open subset of $\mathbb{R}^m$, is such that $Df(p)$ has rank $k$ for all $p\in U$. Show that for each $p\in M$, there exists a neighborhood $V$ of ...
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### Explain the equation of the tangent plane?

So I'm doing my assignment to calculate the equation of a tangent plane at some point. I stumble upon a page on the web that says that an equation of a tangent plane to the surface $z=f(x, y)$ at the ...