Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 5

Let \(\mathbf{x}=\left(x_{1}, \ldots, x_{m}\right)\) be coordinates in a neighborhood of \(\mathbf{x} \in M\), and \(\mathbf{y}=\left(y_{1} \ldots, y_{n}\right)\) be coordinates in a neighborhood of \(\mathrm{y} \in N\), Let \(\xi\) be the set of components of the vector \(v_{\text {, and }}\) \(\eta\) the set of components of the vector \(f_{* x}\). Show that $$ \boldsymbol{\eta}=\frac{\partial \mathbf{y}}{\partial \mathbf{x}} \xi \text {. i.e.. } n_{i}=\sum_{j} \frac{\partial y_{i}}{\partial x_{j}} \xi_{j} . $$ Taking the union of the mappings \(f_{e x}\) for all \(x\), we get a mapping of the whole tangent bundle $$ f_{*}: T M \Rightarrow T N \quad f_{*} v=f_{* x^{v}} \text { for } v \in T M_{*} $$

Short Answer

Expert verified
#Answer# The components of the pushforward of a tangent vector are given by the multiplication of the tangent vector's components by the Jacobian matrix (partial derivatives) of the coordinate mappings. In other words, if we have a tangent vector \(v\) with components \(\xi_j\) and its pushforward \(f_{*}v\) with components \(\eta_i\), the components of the pushforward are related by: $$ \eta_i = \sum_j \frac{\partial y_i}{\partial x_j} \xi_j. $$

Step by step solution

01

Definition of the pushforward

Given a smooth map \(f: M \to N\), the pushforward \(f_{*x}: T_xM \to T_{f(x)}N\) is a linear map between tangent spaces. This means that for a given tangent vector \(v\) at point \(x\), we can find the tangent vector \(f_{*}v\) at the point \(y=f(x)\) in the tangent space of N. Now, we want to show that the components of the pushforward of the vector are given by the multiplication of the tangent vector's components by the Jacobian matrix (partial derivatives) of the coordinate mappings: $$ \eta_i = \sum_j \frac{\partial y_i}{\partial x_j} \xi_j, $$ where \(\eta\) is the set of components of the vector \(f_{*}v\), and \(\xi\) is the set of components of the vector \(v\).
02

Calculate the pushforward of a tangent vector

Let \(x^i\) be local coordinates on M and \(y^i\) be local coordinates on N around points x and y respectively. Let \(\frac{\partial}{\partial x^j}\) and \(\frac{\partial}{\partial y^i}\) denote basis vectors for the tangent spaces. Now, we have a tangent vector \(v = \sum_j \xi_j \frac{\partial}{\partial x^j}\) at the point x. We want to find the pushforward of this tangent vector \(f_{*}v\), which is a tangent vector at the point y. Using the linearity of the pushforward, we have: $$ f_{*}v = \sum_j \xi_j f_{*}\frac{\partial}{\partial x^j}. $$
03

Calculate the component of the pushforwarded tangent vector

To find the ith component of the tangent vector \(f_{*}v\) in the y-coordinates, we can use the transformation of the coordinates: $$ \eta_i = \left(f_{*}v\right)^i = \sum_j \xi_j \left(f_{*}\frac{\partial}{\partial x^j}\right)^i = \sum_j \xi_j \frac{\partial y^i}{\partial x^j}, $$ This proves the claim that the components of the pushforward of the vector are given by the multiplication of the tangent vector's components by the Jacobian matrix (partial derivatives) of the coordinate mappings: $$ \eta_i = \sum_j \frac{\partial y_i}{\partial x_j} \xi_j. $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Extend Noether's theorem to non-autonomous lagrangian systems. Hint. Let \(M_{1}=M \times \mathbb{R}\) be the extended configuration space (the direct product of the configuration manifold \(M\) with the time axis \(\mathbb{R}\) ). Define a function \(L_{1}: T M_{1} \rightarrow \mathbb{R}\) by $$ L \frac{d t}{d z} \text { : } $$ i.e., in lucal coordinates \(q, t\) on \(M_{1}\) we define it by the formula $$ L_{1}\left(\mathbf{q}, t, \frac{d \mathbf{q}}{d \tau}, \frac{d t}{d \tau}\right)=L\left(\mathbf{q}, \frac{d \mathbf{q} / d \tau}{d t, d \tau}, t\right) \frac{d t}{d \tau} $$ We apply Noether's theorem to the lagrangian system \(\left(M_{2}, L_{1}\right)\). If \(L_{1}\) admits the transformations \(h^{s}: M_{1} \rightarrow M_{1}\). we obtain a first integral \(l_{1}: T M_{1} \rightarrow \mathbb{R}\). Since \(\int L d t=\int L_{1} d \tau_{\text {. this reduces }}\) to a first integral \(I: T M \times\) of \(\rightarrow \mathbb{R}\) of the viginal system. If. in local coordinates \((\boldsymbol{q}, t)\) on \(M_{1}\), we have \(I_{1}=I_{1}(\mathbf{q}, t, d \mathbf{q} / d \tau, d l / d \tau)\), then \(I(\mathbf{q}, \mathbf{q}, t)=l_{1}\left(\mathbf{q}, I_{1}, \dot{\mathbf{q}}, 1\right)\). In particular, if \(L\) does not depend on time, \(L_{1}\) admits translations along time, \(h^{s}(q, t)=\) (q. \(t+5\) ). The corresponding first integral \(l\) is the energy integral.

A rod of weight \(P\), tilted at an angle of \(60^{\circ}\) to the plane of a table, begins to fall with initial velocity zero (Figure 74). Find the constraint force of the table at the initial moment, considering the table as (a) absolutely smooth and (b) absolutely rough. (In the first case, the holonomic constraint holds the end of the rod on the plane of the table, and in the second case, at a given point.)

Show that the vector \(f_{* x} v\) does not depend on the curve \(\boldsymbol{\varphi}\), but only on the vector \(\boldsymbol{v}\).

Suppose that a rigid body is moving under its own inertia. Show that its center of mass moves linearly and uniformly. If the center of mass is at rest, then the angular momentum with respect to it is conserved.

Suppose that a particle moves in the field of the uniform helical line \(x=\cos \varphi\). \(y=\sin \varphi_{z} z=c \varphi\). Find the law of conservation corresponding to this helical symmetry.
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Rotational Dynamics

Read Explanation

Magnetism

Read Explanation

Physical Quantities And Units

Read Explanation

Fields in Physics

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free