u/Next-Natural-675

It’s now October, and the next five days are gonna be over 90 degrees. It’s in the middle of fall and it’s gonna be hotter than it ever was during the entirety of summer. 🔥

Up until integrals I feel like us math students have been given a straightforward handbook on how to proceed with solving equations, for example: product rules, quotient rules, constant rules, when you are given this equation you use that method, when you are given that equation you use that method etc..

With integrals I feel like it is different. Now it's more like: if you are given something that looks sorta like it could be this then you try this method, if you are given something that may look like that sort of integral then you use that method..

Is there not a surefire procedure to approach any given type of integral, simply by observing the integral? Do we need to use trial and error for the more abstract integrals to find the solution?

From google I read that we can calculate the enclosed charge if we know the enclosed surface flux, and if we know the enclosed charge we can calculate the surface flux.

But we don’t have a device that measures the enclosed surface flux. We have a flux meter for a magnetic field, but not for the electric field. So we never start off with knowing the surface flux.

And if we know the enclosed charge, what use is calculating the flux?

The math gives zero but I'm looking for an intuitive explanation. I see from another reddit post that if there was curl, then a path will always loop back around to the same point, and that can't be possible because the gradient always points towards a point of higher elevation and that would be contradictory. But can't the curl be reversed mid path, bc the curl is a single value at every point, then the path won't loop back around and you won't end up with a contradiction where the elevation increases towards the lowest point, the point of origin?

This textbook says: "The integral around a closed path is evidently zero (for then ra = rb):  E · dl = 0, (2.19) and hence, applying Stokes’ theorem, ∇ × E = 0."

What I understand is that the line integral is the integral of the dot product of dr and the electric field integrated with respect to dr, where dr is the change in distance to origin r. This equals the integration of curl over the surface with respect to dA. So if the curl is zero everywhere, then the line integral must be zero. What is the proof that if the line integral is zero, the curl must be zero?

Stokes theorem doesn't seem to imply that

This textbook says the divergence is a measure of how much the vector "spreads" out at a point in the vector field. Does this mean its a measure of the difference in length between two vectors next to each other? But wouldn't that be the difference in length of two vectors divided by the distance between the two vectors' starting points? Which is given by the pythagorean theorem?

I heard this is an abuse of notation, but also that there is a rigorous proof for this. I would like to know what that proof is.

I’ve tried spamming the abuse of notation symbol onto the sum of notations using matrix multiplication. I need the divergence curl to compute it.

Sorry that was word scramble to get past the screening bot

So a 3x3 linear transformation matrix is a tensor, if I'm understanding correctly. What the matrix does is transform some vector into another vector.

This textbook I am reading gives this complicated solution to the rotation of coordinates of this 3x3 linear transformation matrix.

But wouldn't it just be equivalent to, when a vector is multiplied by the matrix, then that transformed vector being transformed by a 3x3 rotation matrix?

This is probably a stupid question, but if gr doesnt work at small scales, isnt it possible that gr is correct, but just only at large distances?
I guess that gets into the philosophy of what makes a valid theory, but if the minimum condition is only that it can be expressed as an equation, a piecewise function type thing would suffice no?

How did Planck go from "The ultraviolet catastrophe can be resolved by making the energy of an electromagnetic wave discrete" to "Any physical entity with one degree of freedom whose "coordinate" is a sinusoidal function of time (i.e., executes simple harmonic oscillations) can possess only total energies E' which satisfy the relation E=nhv"?

From what I understand, in a metallic blackbody box, there are three dimensions (three pairs of opposite walls) that harbor each component of a 3d electromagnetic wave. Doing calculations to determine the frequency of the 3d wave tells us that the amount of possible frequencies of this 3d wave within a fixed range of frequencies increases as the frequency increases proportional to the square of the frequency. Therefore, the total energy within a given fixed range of frequencies increases exponentially.

But if the radiation is actually composed of one dimensional waves travelling back and forth across opposite walls, why can't we just look at the total energy within a given fixed range of frequencies for one dimension (or one pair of walls), which is constant regardless of which frequencies the frequency range is at, (due to the equipartition theorem, considering the pair of opposite walls are a single system), then we just multiply that energy by three for all three pairs of walls, or dimensions? Then the energy within a freq range doesn't increase exponentially with the frequency.

Then if you heat up one pair of opposite walls, that component wave will have more average energy than the other component waves, assuming the waves are perpendicular to each wall, not bouncing off the other walls.

Or does it not matter because if you combine each component to get the three dimensional wave, the average energy of each frequency is the same?

In hydrogen, I thought the s-orbital was time independent and the probability was never negative anywhere. So the wavefunction is still, and positive everywhere right? Then how can we get antibonding molecular orbitals? This text says that if the wavefunction of the electron in one of the atoms in a dihydrogen molecule has the opposite sign of the electron in the other atom, they're out of phase and cancel out. But how can one wavefunction be the opposite sign of the other, if they are all positive?

If I urinate onto some fecal matter in a public porta potty, does the fecal matter diffuse up the stream fast enough to make contact with me before I am finished urinating