7
votes
Accepted
Vector addition for differentials in the context of electric potential
I think that your professor is showing the differential vector for infinestimal change in each coordinate component.
The diagrams correspond to cartesian, cylindrical and spherical coordinate systems ...
5
votes
What does an upside down delta mean - covariant vectors?
That is the Nabla, and it just means the covariant vector made of space derivatives.
Edit:
$$\vec\nabla f=
\begin{pmatrix}
\dfrac{\partial f}{\partial x}\\
\dfrac{\partial f}{\partial y}\\
\dfrac{\...
- 5,133
5
votes
Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
4
votes
Given a wave that follows these two conditions, is there some valid and non-trivial physical interpretation to it?
By Helmholtz's decomposition theorem if $\nabla \cdot \vec{u}=0$ and $\nabla\times\vec{u}=0$ in all space then $\vec{u}=0$.
- 14.3k
4
votes
Accepted
How does the $\not{\partial}$ work in the Dirac Lagrangian?
The $\equiv$ symbol in eq. (2.142) allegedly means equality modulo total divergence terms (which don't change the Euler-Lagrange (EL) equations).
- 188k
3
votes
Partial derivatives vs Covariant derivatives in polar coordinates
As OP correctly points out connections introduce a concept of differentiation of tensor fields or more in general of sections of vector bundles that takes into account how the bases of the fibers ...
- 1,607
3
votes
Accepted
Why can the dot product of two vectors be expressed as a differential?
I would imagine that the simplest way to show this is to note that the position vector $\mathbf x$ can be expressed in either basis:
$$x'^j \hat e_j' = \mathbf x = x^i \hat e_i$$
A given set of ...
- 63.6k
3
votes
Difference and meaning of index the derivative operator
You can eventually (if you need to) learn a more rigorous treatment later, so let me instead provide a cookbook approach:
An object with an open index means that its value changes when the observer ...
- 769
3
votes
Vector addition for differentials in the context of electric potential
The infinitesimal displacement $d\vec{s}$ is derived from $\vec{s}$.
In cartesian coordinates :
$\vec{s}=x\vec{i}+y\vec{j}+z\vec{k}$
$$d\vec{s}=dx\vec{i}+xd\vec{i}+dy\vec{j}+yd\vec{j}+dz\vec{k}+zd\vec{...
- 186
3
votes
Accepted
Why the $\Delta$ in the definition of pressure? (fluid mechanics)
The $\Delta$ is there to indicate that it is the "bit" of the total force on the boundary due to the pressure pushing on the small area $\Delta S$.
$P= \Delta F/\Delta S$ is better written ...
- 48.2k
2
votes
Why the $\Delta$ in the definition of pressure? (fluid mechanics)
Pressure is the amount of force applied perpendicular to the surface of an object per unit area.
A problem arises when the surface is not flat ie curved.
So what is done is to take an infinitesimally ...
- 87.1k
1
vote
Accepted
Invariance of d'Alembertian for Lorentz boost in arbitrary directions
I have not been able to have a thorough look at this, but maybe this could be the problem. In your first equation with $\nabla_{\mathbf x'}$ you calculate $\frac{\partial t}{\partial\mathbf x'}$. To ...
1
vote
Accepted
Hermitian Conjugate Terms in Lagrangians
Under Hermitian conjugation,
$$
\phi^{\dagger} \phi \mapsto \phi^{\dagger} \phi ,\implies \\
i\phi^{\dagger} \partial_\mu \phi \mapsto i\phi^{\dagger} \partial_\mu \phi -i\partial_\mu ( \phi^{\...
- 55.7k
1
vote
What do we get on differentiating the instantaneous displacement function?
Displacement is defined as difference between position vectors :
$$ \vec s = \vec r - \vec r~' \tag 1$$
Differentiating that we get :
$$
\frac {d\vec s}{dt} = \frac {d(\vec r - \vec r~')}{dt} = \...
- 10.3k
1
vote
Does $δS = 0$ mean that "the small changes in the actions equal to zero"?
Indeed the diagram you show is not correct, since you are interpreting variations of the action along the curve that minimizes it (sorry if I misunderstood the picture), and that is not the point of ...
- 549
1
vote
Equating 2 sides of EFE
First, note that for scalar functions, the covariant derivative reduces to the partial derivative. So for scalar functions, it is true that if the covariant derivative is zero at a point, then the ...
- 42.9k
1
vote
Chain rule when the intermediary variable might be equal to zero
Recall the definition of velocity in 1D.
$$v(t)=\lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}=\lim_{t\to t_0}\frac{x(t)-x(t_0)}{t-t_0}=\frac{dx}{dt}$$
From this definition, you can recall that the ...
1
vote
Why can the dot product of two vectors be expressed as a differential?
First observe, that for any matrix, we can pick up its $i,j$ entry by applying it first to a column vector that is zero apart from $1$ at its $j$'th position and then dotting it into a similar vector ...
- 1,256
1
vote
Difference and meaning of index the derivative operator
Here is a brief summary:
In (relativistic) physics, it is standard to adorn a local coordinate $x^{\mu}$ with a superindex.
We define a shorthand notation for the partial derivative $\partial_{\mu}:=...
- 188k
1
vote
Vector addition for differentials in the context of electric potential
The diagrams are slightly misleading, because the infinitesimal changes in angular quantities are shown as being quite large. It may be more helpful to draw separate diagrams showing how $\vec{s}$ ...
- 1,217
1
vote
Covariant derivative of gauge theory in curved space
It depends on what sort of field $\phi$ is. If $\phi$ is a scalar field, you can use the ordinary derivatove, but if $\phi$ is a Dirac spinor, for instance, then you will need to include the spin ...
- 48.2k
1
vote
Accepted
Calculating the Berry potential: Questions
Yes
To quote the relevant passage of Sakurai:
Let us continue our discussion by taking as an example, the problem of the spin in a magnetic field mentioned earlier (spin magnitude $S$, and magnetic ...
- 63.6k
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