@bcrowell - I cannot say anything about the WP conformal diagram (it looks flat out wrong at first glance, but I'll get back to you). As for the diagram in #3, it is common practice to draw the center as a vertical line (since the center is supposed to be timelike). But if you actually perform a...
I'm not sure what is part of Calculus II (Non-US bachelors, sorry). But in any case, if using Cartesian coordinates, having your origin at the top of the spherical cap (the north pole of the sphere) gives the bounds as x = 0 to r, y = 0 to r and z = 0 to -h, with the constraint that x2 + y2 + z2...
Magnitude is the absolute value, so, yes, it will always be increasing.
If your answer was "it depends on relativity, being as its a vector quantity (i.e. up can be negative or down can be negative)", it's absolutely right.
But if instead your answer was "it depends on relativity, being as its...
Yes, that is correct.
It's possible that your teacher was thinking about the magnitude of the velocity (also known as speed) which does increase as the object starts falling downwards.
The boldface part is not right. What you have in the first line is
V_{1} = \frac{1}{T_{1}-10} - 4.5
and the second line doesn't agree with this. It should read
\frac{1}{T_{1}} = \frac{1}{T_{1}-10} - 4.5
Yes, that is correct.
I think instead of ft/s, you should stick with miles per hour. Then, you will have two equations in two variables! Do you see it?
Yes, here you can use velocity and speed interchangeably.
Notice how the question says "When you increase your speed by 4.5 mi/h" ... it means V2 = V1 + 4.5. How do the equations look now?
It seems a popular assumption -- "Consider a black hole that forms from collapse of some pure state" (quoted from the AMPS paper). I don't see an obvious reason for this though. Can someone explain this or direct me to one?
I believe they are trying to tell you to have both S_1 and S_2 in the same basis, i.e, either choose S_1 to lie along z-axis and S_2 to lie theta away from it, or choose S_2 to lie along z-axis and S_1 to lie -theta away from it.
Since they already chose the first of these options, the hint is...
Your intuition(?) is correct here. One of the ways to produce a constant electric field is to have two very long charged plates (or, sitting in the middle of two charged plates placed very close to each other). The charge density on each plate to produce such an electric field is given by your...
Yes! If you were to draw the electric field lines, they would be equally spaced parallel lines, parallel to the z axis, pointing from negative to positive direction (or positive to negative, I'm not very sure).
And no problem!
Near wherever you found that expression of radial electric field, should be an expression for electric field in cartesian coordinates. If not, Wikipedia goes in detail about gradients in different coordinates system. If that article is too loud, here is gradient in the three-dimensional...
Yes, with a slight modification. Your electric potential is written in cartesian coordinates, so it'll be easier to have the gradient (and the electric field) in cartesian coordinates also.
How are electric field (the thing you want) related to electric potential (the thing you have)? Before I give details, I'll give a short answer: Electric field is the negative gradient of electric potential. You can find the definition of gradient in case you don't know it, in your...
Your confusion is justified. I was not doing the calculation correctly.
But the idea is right. Your professor has explained it mathematically already, my earlier post (below) explained it in English. I will try to explain it more clearly later tonight (unless you figure out that 4a_{10} =...