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 attempt
What I have tried is to express this...
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...
I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.
Tensors. As...