0
votes
1answer
29 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
98 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
21 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
45 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
36 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
50 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
25 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
64 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
60 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
290 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
38 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
19 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
64 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
30 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
64 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 ...
0
votes
0answers
11 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
37 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
1answer
71 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
46 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
30 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
91 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
47 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
23 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
82 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
50 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
54 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
38 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
79 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
54 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
87 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
48 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
50 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
78 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 ...
3
votes
2answers
191 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
46 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
31 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
215 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
550 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
94 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
40 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 ...
2
votes
1answer
44 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
43 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 ...
7
votes
2answers
769 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
71 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
96 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 ...