Muphrid

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May 19	revised	In coordinate-free relativity, how do we define a vector? corrected a slight error in wording
May 19	comment	In coordinate-free relativity, how do we define a vector? @BenCrowell I've added a section to my answer that may be more relevant.
May 19	revised	In coordinate-free relativity, how do we define a vector? added a section more relevant to the actual intent of the question
May 19	answered	In coordinate-free relativity, how do we define a vector?
May 18	comment	Why distinguish between row and column vectors? I don't see how you need tangent vectors for hypersurfaces. The curves used to define tangent vectors are just maps from the reals to points on the manifold. Arbitrary hypersurfaces are just maps from $N-1$ reals to points in the manifold. Your observation about the necessity of a volume form seems reasonable. What kinds of manifolds lack volume forms?
May 18	comment	Why distinguish between row and column vectors? @Christoph Yes, tangent vectors can be defined through a family of curves through a point that all have the same direction. Is there something wrong with defining cotangent vectors via an equivalence class of hypersurfaces, containing a particular point, that all locally have the same linear approximation?
May 18	comment	Why distinguish between row and column vectors? Yes, that is true. I think what is more at issue is the choice that "vectors" mean tangent vectors and "dual vectors" mean cotangent vectors. The choice of what is original and what is dual is arbitrary. I agree that means we can't go in circles with our definitions, however.
May 18	answered	Why distinguish between row and column vectors?
May 18	answered	Can general relativity be completely described as a field in a flat space?
May 17	answered	Vector cross product of $\mathbf{r}$ and $\ddot{\mathbf{r}}$ in polar coordinates
May 14	comment	Electric field and electric potential of a point charge in 2D and 1D Given a little tensor machinery, you can give $\nabla$ its own vector Green's function also. Much less circuitous, in my mind, than having to backtrack to potentials.
May 14	answered	What do people actually mean by “rolling without slipping”?
May 13	comment	Geodesic equations Why did you choose the Euler-Lagrange equations instead, of, say, the geodesic equation for general relativity?
May 9	comment	Magnetic Field versus Line Charge The EM field is a field of oriented planes through spacetime. These planes are spanned by (1) the direction of the current (whether through space or time) and (2) the direction from the observer to the current (in space). Electric field planes come from currents that only go through time (such currents represent static charges), but when those currents have spatial components, they generate magnetic planes also.
May 7	comment	Vector potential @Mike Yeah,I had to derive it from scratch because I couldn't find it anywhere, just to check the sign of the cross product.
May 7	comment	Is there a momentum for charge? The center of charge you've defined is something like a normalized dipole moment. The "charge-momentum" is the same as a current.
May 6	answered	Vector potential
May 5	revised	EM Fields in a Rotating Frame of Reference corrected an error
May 5	answered	EM Fields in a Rotating Frame of Reference
May 3	comment	Relativistic origin of magnetic field Think about conservation of momentum: the $x, y, z$ components of momentum are all individually conserved, but you don't expect them to stay the same when you rotate a coordinate system. Charge and current are the same way, but with Lorentz boosts--there's a basic idea of conservation, but that doesn't apply between frames, only within individual frames. The charge on a neutral wire is zero because that's what neutral means--the charge and current densities must be constructed to make it so. Conserved quantities correspond to symmetries; it has nothing to do with being variable.