0
votes
0answers
34 views

Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
0
votes
1answer
36 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
0
votes
1answer
45 views

Flow of a differential equation over what interval

Let $\dot{x}=x^2$. Over what interval is the flow defined? I can see that the solution is of the initial value problem $\dot{x}=x^2$, $x(0)=x_0\ $ is $$ x(t)=\frac{x_0}{1-x_0\cdot t}$$ and that it ...
1
vote
2answers
44 views

What does it mean to take the cross product of velocity and acceleration?

This is from a practice question I am working on. The osculating plane to the curve given by the vector valued function $r(t) =\langle\cos(t), (t-1)^2, -\sin(t)\rangle$ at the point corresponding ...
2
votes
1answer
49 views

Derivative of Quadratic Form as a Linear Approximation

I'm trying to find the derivative of the $quadratic$ form, for a $symmetric$ $n$ by $n$ matrix A and $ x \in \mathbb{R}^n $, $$ f(x) = x^tAx $$ such that the derivative is a linear map from $ ...
3
votes
2answers
72 views

Evaluating $\int_0^{2\pi} \sqrt{1+\cos(t)} \ \mathrm{d}t$

Context: I'm trying to evaluate the total length of the following curve: $\gamma: (0,2\pi) \to \mathbb{R^2}, \gamma(t)=\bigg(t+\sin(t), 1-\cos(t)\bigg)$. $$\underline{\text{My working}}$$ ...
1
vote
3answers
50 views

How to define derivative in Minkowski space

My understanding of derivative is like this: it is the unique linear mapping that sends the difference in $x$ to the difference in $f(x)$ when the difference in $x$ is small. To put it more ...
3
votes
1answer
157 views

calculation of normal derivative

Suppose $\Omega$ is a bounded region in the plane $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Suppose $u$ is a smooth function in $\Omega$. I want to calculate ...
1
vote
0answers
49 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
1
vote
1answer
54 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
0
votes
1answer
68 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
0
votes
0answers
25 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
1
vote
1answer
45 views

The level set of a smooth function

Let $f$ be a smooth function on a manifold $M$. Fix a point $p\in M$ and let $df\in T^\ast_pM$ be the differential of $f$ at $p$. I read that the subspace of $T_pM$ of vectors $X$ such that $df(X)=0$ ...
2
votes
0answers
36 views

Connected Sum: Actual Example

Suppose I have two tori as in the image: I have parameterizations of each torus and I want to form a nice $C^\infty$ connected sum. How do I do this? I know the theory, but not the practice. How do ...
1
vote
1answer
100 views

The relation between geodesics and distances on a Riemannian manifold

My question is about computing the distance between two points in a Riemannian manifold. Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex. Let ...
0
votes
1answer
31 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
3
votes
2answers
117 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
1
vote
0answers
31 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
3
votes
0answers
48 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
2
votes
2answers
37 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
3
votes
2answers
85 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
2
votes
0answers
26 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
1
vote
3answers
516 views

maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: ...
2
votes
2answers
263 views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
1
vote
2answers
291 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
0
votes
1answer
42 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
1
vote
0answers
20 views

Any elementary derivation of the Pfaff integrability condition?

Suppose in $\mathbf{R}^N$ we have a one-form field, $ \theta = \sum_{i=1}^N \theta_i d x_i $. The Pfaff integrability condition is $d \theta \wedge \theta = 0$. Is it possible to give an ...
3
votes
1answer
67 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he ...
0
votes
0answers
31 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
7
votes
1answer
68 views

Translate a vector field

Imagine that you have a vector field $A = \frac{A_0}{r} e_{\theta}$ in cylindrical coordinates, where $A_0 \in \mathbb{R}$. Now you translate your coordinate system in $e_x$ direction by $x \mapsto x ...
1
vote
0answers
13 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
1
vote
0answers
38 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...
0
votes
2answers
71 views

How to find the equation of the normal line to the surface S

How to find the equation of the normal line to the surface $S$: $$f(u,v)=(2u-v,u^2+v^2,u^3-v^3)$$ at the point $M(3,5,7)$? Could someone post the complete solution?
1
vote
2answers
88 views

Radius of curvature for the plane curve $x^3 + y^3 = 12xy$.

Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.
0
votes
1answer
15 views

mapping between differential forms and its property

I am trying to prove the following property of the map between differential forms: (Spivak's book ''Calculus on manifolds'' p.91) $$f^{\star}\;\Lambda^{k}(\mathbb{R}^{m}_{f(p)})\to ...
0
votes
2answers
48 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
1
vote
1answer
31 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
43 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
269 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
49 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
24 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
92 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
29 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
59 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
58 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
41 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 ...
2
votes
2answers
91 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
57 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 ...
3
votes
2answers
109 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 ...