I have seen that there are two different formulas that we can use when calculating the propagation of uncertainty in a measurement. If ##X=f(A, B, C, \ldots)## is the quantity whose uncertainty we want to estimate, which depends on the quantities ##A,B,C,...##, then we could calculate the...
I too think that the demonstration should be shorter than what I have tried, but I don't find the problem...
By the way, the components of the transition matrix ##(C^i_j)## of my notation are the terms ## \frac{\partial x^i}{\partial x'^j}## of your notation, aren't they?
In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation)
##
\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}
##
I'm trying to prove that this covariant...
According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of...
Well, with the answers I have read, I think I have managed to clarify myself with this topic (also, looking at vanhees71's articles, it seems that the topic is extensive). Thank you very much :)
Rather than a book, I should have said notes, it's a text written by the professors for the Quantum Mechanics course I am taking (and not a very good one, by the way). I'm sorry, I don't have more information...
An electron beam with the spin state ## |\psi\rangle = \frac{1}{\sqrt{3}}|+\rangle+\sqrt{\frac{2}{3}}|-\rangle##, where ##\{|+\rangle,|-\rangle\}## is the eigenstates of ##\hat S_z##, passes through a Stern-Gerlach device with the magnetic field oriented in the ##Z## axis. Afterwards, it goes...
According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is:
##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0##
Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...