Well you have compactified some steps (for example i would like to see the representation of the radial field in cartesian coordinates and the derivatives of ##r=\sqrt{\sum x_i^2}## with respect to each cartesian coordinate) but ok i am more than surprised that it can be done in less than half a...
Like the others i dont see anything wrong in your work. Maybe it is kind of simple because it is done in spherical coordinates. Try to do it in cartesian coordinates i think it would be one page long or even more!.
First use the pythagorean theorem to find the hypotenuse. Then use the definition of sin cos and tan on an angle of a right triangle. For example it is according to the definition that ##\sin\phi=\frac{a}{\text{hypotenuse}}##
You mean because centripetal acceleration "suddenly" appears there? As far as i know acceleration need not be continuous, its velocity that is usually continuous.
Q2 is straightforward application of conservation of momentum.
Q1 both a) and b) are about application of the principle of conservation of energy.
I am sorry i am not allowed to say more according to the rules of PF you got to show us your best attempt at solution, which means you have to work...
@PeroK post #4 might seem funny but he says a lot, its the force from the floor/seat that is responsible for this effect. Forget my post about fictitious forces.
There are two answers in your question depending in which frame of reference we ll answer.
In the accelerating frame of reference of the rocket, there is a fictitious force that presses the man down. Maybe its not the right time to be introduced to fictitious forces but you can google about...
There is indeed a problem with #2, you have to prove that it holds in any base but i dont understand the problem with #1, how are you supposed to work with vectors if you dont adapt a vector space(R^3 or R^2)?
You are right that the proof with u and v is more general more complete i would say.
I think in ancient times, electricity and magnetism effects and phenomena though they were known, they were viewed as toy-like and ancient civilization just didnt put any research on these phenomena. They just couldn't imagine that one could make an electromagnetic engine e.g an electric motor...
Its from the energy the water has due to its pressure (static pressure) which in turns comes from gravitational potential energy or from some pump that is located somewhere and increases the pressure of the water.
You shouldn't cross this in my opinion it was correct. It is just that it turns out that ##t_2=0## and ##d_2=1## which means that he has to run the second mile with infinite speed ##\frac{d_2}{t_2}=\infty##. But the problem asks if it is possible, so i guess the answer here is NO it isnt...
I have to look it up abit more thoroughsly (give me time to think heh ) but we can agree for now that your equation is correct in the case of non time varying currents.
It doesnt seem to be correct in the case of time varying currents cause then the well known expression for A contains the...
Are you sure this is the equation derived by Feynman? This equation obviously fails if ##\mathbf{B}=constant\neq 0## cause it gives ##\nabla\times \mathbf{B}=0## and hence ##\mathbf{A}=0##.
I have in mind a slightly different equation which i derive from Helmholtz decomposition theorem which is...
This formula you wrote $$f(x)=Q(x)(x-1)(x+1)+R(x)$$ says a lot if you know how to interpret it. Since (x-1) divides f(x) what can you say about whether (x-1) divides R(x)?
I believe this and together with the answer to the question of @haruspex will get you to the answer of the question.