The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move...
When we solve Euler's differential equations for rigid bodies we find the angular acceleration ##\dot{\boldsymbol\omega}## and then the angular velocity ##\boldsymbol\omega##. Integrating ##\boldsymbol\omega## is less straightforward, so we start from a representation of the attitude, take its...
1. Homework Statement
Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$
\int_{\mathbb{R}^n} d\omega = 0.
$$
2. Homework Equations
$$\oint_{\partial K} \omega = \int_K d\omega$$
3. The Attempt at a...
In the exercises on differential forms I often find expressions such as $$
\omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz
$$ but this is only correct if we're in "flat" space, right?
In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...
Let ##M## be an ##n##-dimensional (smooth) manifold and ##(U,\phi)## a chart for it. Then ##\phi## is a function from an open of ##M## to an open of ##\mathbb{R}^n##. The book I'm reading claims that coordinates, say, ##x^1,\ldots,x^n## are not really functions from ##U## to ##\mathbb{R}##, but...
I'm reading Scheck's book about Mechanics and it says that Newton's first law is not redundant as it defines what an inertial system is. My problem is that we could say the same about Newton's second law. Indeed, Newton's second law is only valid, in general, for inertial systems, so it also...
Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...
<Moderator's note: Moved from another forum.>
The book I'm reading says that ##\star \sigma = 1## and ##\star 1 = \sigma##, but I'm not sure about the last one. The space is ##V = \mathbb{M}^4## and we choose the canonical base ##e_0,e_1,e_2,e_3##. This means that ##g_{ij} =...
I've just learned about the covariant derivatives (##\nabla_i## and ##\delta/\delta t##) and I have a doubt.
We should be able to say that $$
J^i = \frac{\delta A^i}{\delta t}
= \frac{\delta^2 V^i}{\delta^2 t}
= \frac{\delta^3 Z^i}{\delta^3 t}
$$ where ##J## is the jolt. This...
I hope you can understand my notation. The Christoffel symbol can be defined through the relation$$
\frac{\partial \pmb{Z}_i} {\partial Z^k} = \Gamma_{ik}^j \pmb{Z}_j
$$ I can solve for the Christoffel symbol this way: $$
\frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m = \Gamma_{ik}^j...
Here's exercise 1 of chapter 2 in Flanders' book.
Let ##L## be an ##n##-dimensional space. For each ##p##-vector ##\alpha\neq0## we let ##M_\alpha## be the subspace of ##L## consisting of all vectors ##\sigma## satisfying ##\alpha\wedge\sigma=0##. Prove that ##\dim(M_\alpha)\leq p##. Prove also...
I'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts.
Let ##\lambda## be a ##p##-vector in ##\bigwedge^p V## and let ##\sigma^1,\ldots,\sigma^n## be a basis of ##V##. There's a unique ##*\lambda## such that, for all ##\mu\in \bigwedge^{n-p}##,$$
\lambda...
The support for ##\LaTeX## is great, but there's just one thing I don't like: if I put a newline after the closing \$$ I get too much vertical space after the rendered part. For instance:
$$(x_1+\ldots+x_p)^n = \sum_{c_1+\ldots+c_p=n} \frac{n!}{c_1!\cdots c_p!}x_1^{c_1}\cdots x_p^{c_p}$$
As you...
I'm learning Differential Geometry (DG) on my own (I need it for robotics). I realized that there are many approaches to DG and one is Cartan's, which is presented in Vargas's book. I think that book is highly opinionated, but I don't know if that's a good or bad thing. Does anyone of you know...
I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining ##||X||## in an ##M^n## by
$$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to...
I'm learning Differential Geometry on my own for my research in ML/AI. I'm reading the book "Gauge fields, knots and gravity" by Baez and Muniain.
An exercise asks to show that "if \phi:M\to N we can push forward a vector field v on M to obtain a vector field (\phi_*v)_q = \phi_*(v_p) whenever...
My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular.
The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...
Hi,
I need to minimize, with respect to \hat{y}(x), the following function:
\tilde{J}_x = \mathbb{E}_{p(x,y)}[(\hat{y}(x)-y)^2] + \nu \mathbb{E}_{p(x,y)}[(\hat{y}(x)-y)tr(\nabla_x^2\hat{y}(x))] + \nu \mathbb{E}_{p(x,y)}[||\nabla_x\hat{y}(x)||^2],
where x is a vector and y a scalar.
I found this...