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) = ...
3
votes
0answers
47 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
1
vote
1answer
54 views

Inverse Function Theorem for Manifolds with Boundary

In Lee SM it is written that the inverse function theorem can fail for manifolds with boundary.As hint it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\to\mathbb{R}^n$ ...
0
votes
0answers
35 views

Covariant Derivatives of contravariant vector in curvilinear coordinates

$$D_mA^p = \partial_mA^p + \Gamma^p_{mn} A^n$$ so $$D_kD_mA^p = D_k(\partial_mA^p + \Gamma^p_{mn} A^n)$$ $$D_kD_mA^p = \partial_k(\partial_mA^p + \Gamma^p_{mn}A^n) + \Gamma^p_{kl}(\partial_mA^l + ...
1
vote
1answer
26 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 ...
0
votes
1answer
41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
0
votes
1answer
40 views

Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
0
votes
1answer
68 views

Finding Arc Length using First Fundamental Form

Let $v=ln(u+\sqrt{u^2+1})+C$ be the curve given on the right helicoid $x=u\cos(v),y=u\sin(v),z=2v$. Calculate the arc lengths of this curve between the points $M_1(0,0)$ and $M_2(1,ln(1+\sqrt{2}))$ ...
1
vote
1answer
87 views

Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d $ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)} $ be a family in $\mathrm{Diff}(\Omega)$ such that $ \phi_0 = id_{\Omega}$ and ...
1
vote
1answer
42 views

A question about covariant derivative (find $D_{v_p}W$)

the quest. on my book is $(y_1,y_2,y_3)\quad R^3\quad coordinate \quad system$ . if $W=y_1y_2^2\frac{\partial}{\partial y_1}+(y_3-y_2^2)\frac{\partial}{\partial y_2}+3y_1\frac{\partial}{\partial ...
2
votes
2answers
61 views

Characterization of differentiability via Lie derivatives

Yesterday I asked this question in MathOverflow but did not receive an answer yet. I want to try my chance here too, since I am in kind of a hurry. Answers will be much appreciated. I intend to ...
0
votes
1answer
27 views

A question about derivative transforms

Firstly I'm not sure "derivative transforms" is the correct statement. There is a question on my book: if $f : \Bbb R^n \mapsto\Bbb R $ and $v_q\in T_q(\Bbb R^n)$ then show that ...
2
votes
1answer
69 views

differential calculus in $\mathbb{R}^\mathbb{N}$?

is it possible to define the derivative of a function of countable variables? I found differential calculus of function with a finite number of variables, or differential calculus in Banach spaces ...
0
votes
1answer
69 views

Prove (local) converse to the implicit function theorem

The implicit function theorem tells us that: Given a level set $M^k = F^{-1}(F(p_0))$ of a smooth function $$F: \mathbb{R}^n \to \mathbb{R}^{n-k},$$ where $\operatorname{rk}{(Df)(p)} = n-k$ for ...
3
votes
1answer
69 views

Multiple variables in derivative and finding the slope

I'm working on slope and geometrical differentiation. The problem is to find the slope of any point on the curve $$\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1$$ I have found its derivative: $\frac{dy}{dx} ...
2
votes
2answers
133 views

shortest distant of curve from origin

what'd be shortest distance of curve from origin(0,0) function is $$ y=\frac{e^x+ e^{-x}}{2} $$ I tried taking some x and y points on curve then using distance formula finding distant then i found ...
1
vote
2answers
70 views

Finding x when slope = 1

I've been working out some problems relating to slope on the points of a curve. I'm having issues with this one: In the curve to which the equation is... $$x^2 + y^2 = 4$$ find the value of $x$ at ...
4
votes
1answer
115 views

Is this really a typo?

Let $U \subseteq \mathbb R^n$ and $F: U \to \mathbb R^m$ a function with coordinate functions $f_i$. My notes say that: If $F$ is differentiable on $U$ the Jacobian of $F$ is defined at each point in ...
1
vote
2answers
87 views

connection along a map

I'm interested in connection along a map between manifolds, i.e. the pullback construction. Let $f: M \rightarrow N$ be a map and $(V,\nabla^V) \rightarrow N$ be a vectorbundle with connection. Then ...
1
vote
1answer
68 views

Does nullity at one point implies nullity everywhere?

Consider the following definition of the derivation. Now, consider a derivation $\delta_{p} :C^{r}(R^{n})\rightarrow R$ , i.e. it is defined on r-times differentiable functions defined on the ...
0
votes
2answers
60 views

Find the maximum rate change.

Find all points from the domain of $$ f(x,y)=e^{x^2-xy-1} $$ in which the function f reaches the maximum rate change (I mean gain/increase) in the direction of x-axis the domain: (am I right?) $$ x ...
3
votes
0answers
73 views

The derivative of a family of flows

If one has a family of flows, can one describe the derivative in the "family" direction? Specifically, let $M$ be a smooth manifold and let $X_{s,t}$ be a 2-parameter family of fields on $M$. That ...
4
votes
1answer
83 views

Complexified tangent vector, complex manifold

Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
1
vote
2answers
291 views

Relationship between second order derivatives and cross derivative of smooth surfaces

Probably a silly question, but I wonder if $z=f(x,y)$ is a smooth surface, and the values of its two second order derivatives $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ ...
0
votes
1answer
88 views

Nondegenerate critical point

I don't understand this part from the book of Zeidler , can someone help me to understand it ? Please Thank you
8
votes
2answers
270 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
0
votes
0answers
74 views

Why Local Minimum is calculated for a derivative function instead of actual function?

In Machine learning regression problem, why the local minimum is computed for a derivative function instead of the actual function? Example: http://en.wikipedia.org/wiki/Gradient_descent The ...
6
votes
1answer
122 views

Compute the differential of a smooth map

Let $S\subseteq \mathbb{R}^3$ be an oriented regular surface and let $N$ be a field of normal unitary vector on $S$. We consider the map $F:S\times \mathbb{R}\rightarrow \mathbb{R}^3$ defined by ...
5
votes
1answer
113 views

For a differentiable map $\Phi$ between manifolds $M$ and $W$, what is $d\Phi?$ (Aubin)

I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this: $(\Phi_*)_P$ is nothing ...
2
votes
1answer
139 views

Equivalent definition of Tangent Spaces

There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ...
2
votes
0answers
83 views

Are there more types of differential in mathematics?

I am familiar with two types of differential normal differential: $$d^2x^a$$ covariant differential: $${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$ (where the covariant differential is broken ...
2
votes
1answer
324 views

Laplacian in $\mathbb{R}^n$ expressed with Christoffel symbols

I thought I'd try my luck with another question, so here it goes: I have to show that if $(y^1,...,y^n):\mathbb R^n \rightarrow \mathbb R^n$ is a diffeomorphism and $f\in C^\infty(\mathbb R^n)$. Then ...
6
votes
2answers
489 views

Why does the condition of a function being differentiable always require an open domain?

Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
1
vote
1answer
286 views

Commutators, and Christoffel symbols in a non holonomic basis

I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors. I need to compute ...
0
votes
1answer
42 views

Hessian equivalence

Let $F: R^n \longrightarrow R$ be twice differentiable and $x,y \in R^n$ with $F(x)=F(y)$. Further let $\phi [0,1] \rightarrow R^n$ be a nice curve with $\phi(0)=x$ and $\phi(1)=y$. If we know that ...
0
votes
1answer
102 views

How to express gradient of a hat function mapped on a triangle as a linear combination of triangle points

I have: a triangle $ABC$ A linear function $f_0(X)$ where: $f_0(A + s(B-A) + t(C-A)) = 1-s-t$ $s,t \in \mathbb{R}$ Can I express $\nabla f_0(x)$ as a linear combination of $A,B,C$? I ...
3
votes
0answers
135 views

Partial derivatives using variables after a transformation

I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this? If it is difficult to find a general expression for ...
4
votes
0answers
38 views

Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define $H^{\alpha}_{ij,k}=e_k\langle ...
3
votes
1answer
134 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
2
votes
3answers
100 views

Calculating a Multivariable derivative.

I'm trying to work through Spivak's Calculus on Manifolds and I've arrived at Differentiation. While I can usually follow his steps, I find myself lost or stuck when I try to do something on my own. ...
6
votes
1answer
259 views

notation of derivation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of derivation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
1
vote
2answers
219 views

Charts, spheres and determinants

Here are some things I don't understand, I would be very grateful for any help! I am trying to find a chart on the unit sphere that preserves area. The most natural map that springs to my mind is ...