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### 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 ...

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### 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 ...

### 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 ...
1 vote
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### 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
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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 ...
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 ...
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 vote

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 $\partial_{\mu}:=... 1 vote ### Vector addition for differentials in the context of electric potential The diagrams are slightly misleading, because the infinitesimal changes in angular quantities are shown as being quite large. It may be more helpful to draw separate diagrams showing how$\vec{s}$... 1 vote ### Covariant derivative of gauge theory in curved space It depends on what sort of field$\phi$is. If$\phi$is a scalar field, you can use the ordinary derivatove, but if$\phi$is a Dirac spinor, for instance, then you will need to include the spin ... 1 vote Accepted ### Calculating the Berry potential: Questions Yes To quote the relevant passage of Sakurai: Let us continue our discussion by taking as an example, the problem of the spin in a magnetic field mentioned earlier (spin magnitude$S\$, and magnetic ...

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