1
vote
1answer
19 views

How to parameterized implicit curve.

How could I parameterize: $$\frac{1}{2}\left(x^2+y^2\right)-\frac{1}{3}x^3=\frac{1}{6}$$ as $x(t)$ and $y(t)$?
2
votes
1answer
39 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
0
votes
0answers
30 views

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?
0
votes
0answers
38 views

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 ...
1
vote
0answers
17 views

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 ...
1
vote
1answer
42 views

Laplace operator in Spherical Coordinates, a formal approach

I'd like to show the well-known formula of the Laplacian operator for euclidean $\mathbb{R}^3$ in spherical coordinates: $$ \Delta U = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial ...
0
votes
0answers
26 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
3
votes
1answer
36 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 ...
2
votes
1answer
47 views

Very basic questions on chain rules and product rules

I have serious gaps in maths and would like to ask some basic questions. I know there is the following chain rule for the first derivative: $$ Dh(x) = Dg(f(x))Df(x)\quad\quad (1) $$ where $h(x) = ...
1
vote
1answer
30 views

Geometrical Calculus - Mini-Max Problem

Two vehicles are heading for a crossroad (point C) and intend to pass straight through. Vehicle A is $100$ km due North travelling at $80$ km/hr towards C Vehicle B is $150$ km due East travelling at ...
1
vote
2answers
62 views

Radius of Curvature

I was asked to show that the expression is constant in a circle : $\dfrac{\left[1+\left(\dfrac{\operatorname d \!y}{\operatorname d \!x}\right)^2\right]^{\frac 3 2}}{\dfrac{\operatorname d ...
3
votes
1answer
47 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
2
votes
2answers
56 views

Find the maximum angle possible

$P$ is a point on the $Y-axis$ . Find the maximum possible value of $\angle APB$ where $A=(1,0)$ and $B=(3,0)$. Here is how I solved the problem. Suppose $P=(0,k)$ . Then using the cosine formula we ...
0
votes
1answer
44 views

implicit function theorem and one to one function

Apply the IFT to show that no $C^1$ function $F:\mathbb{R}^2 \to \mathbb{R}$ can be one to one near any points of its domain. So I know that the theorem says that if we have a point $(a,b)$ such that ...
0
votes
1answer
44 views

All points at which the surfaces $x^2+y^2+z^2-1=0$ and $x^2+y^2-z^2-2y=0$ are intersect orthogonally

$f: x^2+y^2+z^2-1=0$ $g: x^2+y^2-z^2-2y=0$ I set these two surfaces equal to each other to solve for the intersection, getting $y=(1-z^2)/2$...then attempted to insert this value of $y$ in terms of ...
1
vote
2answers
75 views

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
2
votes
2answers
142 views

What does $dx$ mean in differential form?

This question relates to this post. From what I know in calculus and standard analysis, strictly speaking, there is no meaning of $dx$. It only makes sense when combining with another $d$, e.g. ...
2
votes
0answers
44 views

Prove composite is smooth given condition on derivative

The problem I am attempting to prove is: Let $f:\mathbb{R}\to\mathbb{R}$ be smooth. Define $\psi$ by $\psi(x)=x^3$ Show $f\circ\psi^{-1}:\mathbb{R}\to\mathbb{R}$ is smooth $\iff\ f^{(n)}(0)=0$ if ...
0
votes
0answers
24 views

Reality condition on a metric

I'm studying a coördinate-transformation on a 2n-dimensional real manifold, where I can locally define the coördinates as $(x^1,...,x^{2n})\in\mathbb{R}^{2n}$, and transform them to: ...
1
vote
2answers
82 views

differential area

I'm just trying to refresh my calculus a bit, I'm stuck on a question and I'd love some insight. A square measures 0.9cm on each side when drawn with a pencil. When traced over with a marker, it ...
0
votes
2answers
233 views

Shell method for calculating volume of solid of revolution - general

Let us have an injective continuous function $f : [a,b] \to [0,c]$ (such that $f(a)=0$ and $f(b)=c$). I want to calculate the volume of solid revolution of $f$ around the $y$ axis. The first method ...
2
votes
0answers
81 views

How is Euler-Lagrange equation used to find optimal solutions in minimizing a function?

How is the Euler-Lagrange equation: $$ L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 $$ used mathematically in finding the optimal solutions of minimising a function? Can someone give me an ...
0
votes
1answer
37 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 ...
1
vote
1answer
39 views

Question on Construction in Spivak's *Calculus on Manifolds*, induced transformations

First I quote the relevant passage (page 89): If we consider now a differentiable function $f : \mathbb R^n \to \mathbb R^m$ we have a linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$. ...
0
votes
2answers
42 views

how to show $\frac{\partial\hat\sigma}{\partial\hat u}\times\frac{\partial\hat\sigma}{\partial\hat v}$ (cross product)

how to show $$\frac{\mathcal{\partial\hat\sigma}}{\partial\hat u}\times\frac{\mathcal{\partial\hat\sigma}}{\partial\hat v}=\left(\frac{\mathcal{\partial u}}{\partial\hat u} \frac{\mathcal{\partial ...
6
votes
2answers
358 views

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 ...
0
votes
2answers
60 views

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 ...
4
votes
1answer
89 views

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 ...
1
vote
1answer
133 views

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 ...
1
vote
1answer
60 views

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 ...
3
votes
0answers
98 views

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 : ...
1
vote
1answer
36 views

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 ...
2
votes
2answers
51 views

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 ...
3
votes
1answer
93 views

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 ...
3
votes
1answer
50 views

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 ...
0
votes
1answer
58 views

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 ...
0
votes
1answer
56 views

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 ...
1
vote
0answers
43 views

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 ...
0
votes
1answer
102 views

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: $$1. \space\space\space\space S(x) = \sum_{n=1}^\infty (n^2 +n)x^n =$$ ...
0
votes
1answer
28 views

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.
0
votes
0answers
149 views

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) ...
1
vote
1answer
98 views

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 ...
1
vote
2answers
63 views

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 ...
0
votes
1answer
63 views

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 ...
0
votes
1answer
70 views

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 ...
1
vote
1answer
67 views

What is meant by “global smooth coordinates”?

On p. 65 of John M. Lee's book Introduction to Smooth Manifolds, we find the exercise Verify that $\tilde{x}, \tilde{y}$ are smooth coordinates on $\mathbb{R}^3$, where $$\tilde{x}=x; \;\;\tilde{y} ...
0
votes
1answer
73 views

Finding critical points

Let $S\subset \Bbb{R^3}$ be the surface given by $x^2/4+y^2/9+z^2=1$. For $p_0=(1,0,0)$, define $f:S \to \Bbb{R} $ by $f(p)=|p-p_0|$. Then how can I find the critical points of $f$? If I use 2 ...
2
votes
1answer
67 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
3
votes
1answer
82 views

How to compute this angle form integral?

Let $\gamma$ be the curve in $\Bbb{R}^2$ given by $x^2/9+y^2/4=1$ with counter-clockwise orientation. Compute $$\int_{\gamma} \frac{-ydx+xdy}{x^2+y^2}$$ I guess that the answer should be $2 \pi$ ...
2
votes
2answers
44 views

how to calculate frenet serre eqautions

how to calculate frenet serre eqations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =(\cos (\frac{s}{\sqrt 2}), \sin (\frac{s}{\sqrt 2}), (\frac{s}{\sqrt 2}))$$ i know the ...