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1. Adding a total derivative to the Lagrangian

I'm sorry, but I'm just not finding it trivial. Could you write it out for me, please?

13. Figuring symmetries of a differential operator from its eigenfunctions

So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...
14. Representing a factorial through its pseudo Z transform

Ok, so I was playing around with some Z transforms. I'm sorry about the long derivation, but I'm a bit unsure of the mathematical rigor, and want to make sure every step is clear. I started with the recurrence relation defining the factorial: $$n!: u_{n+1}=(n+1)u_n=u_n+nu_n$$ $$u_0 = 1$$...
15. Creating intuition about Laplace & Fourier transforms

I had thought about how the fact that the norm of the basis elements isn't really defined since the integral doesn't converge, but didn't really reach any conclusion... How can you justify integrating over a finite interval? You mention the Hilbert basis won't guarantee pointwise convergence...
16. Creating intuition about Laplace & Fourier transforms

Hey everyone, I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, I don't have any technical issue with that. My only problem is that some...
17. Confused about the Equivalence Principle and Inertial Reference Frames

Ok, Thank you very much, that explains why the same designation is used. Cheers
18. Confused about the Equivalence Principle and Inertial Reference Frames

Ok, I understand now. But why call them both IRF's, if they're clearly different concepts?
19. Confused about the Equivalence Principle and Inertial Reference Frames

Ok, so you agree that the IRF of GR does not coincide with the same concept in classical mechanics, right?
20. Confused about the Equivalence Principle and Inertial Reference Frames

Hey everyone, I started reading up on GR a couple of days ago, and I'm somewhat stuck on the concept of a free-falling IRF. I understand that an observer on a free-falling small spaceship would experience the laws of physics in a rather simple form, eliminating the need for a force of gravity...
21. Proving that a solution to an IVP is unique and infinitely differentiable

Exactly, that's what I was trying to say with recursively differentiating the original equation to obtain an equation for the nth derivative. But thank you, still ;) Cheers
22. Move 'dx' from 'dy/dx' to the other side, then integrate both sides

I don't know anything about that particular book, but any introductory ODE textbook should suit you fine. There's not really that much to know about separable equations, just a couple of tricks to recognize if a certain equation can be made separable under a certain substitution, or rearranging...
23. Move 'dx' from 'dy/dx' to the other side, then integrate both sides

It's not an example of bad mathematics... There is a formal theory of differential forms, and bringing the dx to the other side is just simply used to express a relation between the differentials... Of course, without resorting to differentials, you can just interpret it as a change of variable...
24. Proving that a solution to an IVP is unique and infinitely differentiable

Thank you for your answer. My teacher's suggestion was to turn the equation into an equivalent system of linear differential equations, and then use Picard's theorem for systems to prove uniqueness. I guess I wasn't supposed to directly invoque the existence and uniqueness theorem for linear...
25. Proving that a solution to an IVP is unique and infinitely differentiable

Homework Statement \frac{d^2y}{dt^2} + t\frac{dy}{dt} + t^3y = e^t;\ \ \ y(0) = 0, \ \ y'(0) = 0 Show that the solution is unique and has derivatives of all orders. Determine the fourth derivative of the solution at t = 0. 2. The attempt at a solution I'm somewhat lost here... Trying to...
26. For what value(s) of x does A^-1 exist.

I don't know if you have learned the concept of determinants, but a matrix A is invertible if and only if its determinant is nonzero. The determinant for 2x2 matrices of the form (a b) (c d) is given by det A = ad - bc Computing the determinant and factoring the quadratic polynomial you'll get...
27. DiffEQs modelling/solving Q

You should write the expression of the volume of the cone as \frac{1}{3}y\pi r^2(y), where r(y)=\frac{y}{2}, according to the data they give you.
28. Is this the right integral set-up to find the volume?

Yes, the inner radius should be 4 - inner curve and the outer radius should be 4 - outer curve. I think your setup is correct.
29. Proving that an integral is a pure imaginary

You're totally right, my mistake. I've already corrected it.
30. Proving that an integral is a pure imaginary

Oh, that makes a lot of sense! So one gets: \int_{\gamma}\ f^*(z)\ f'(z)\ dz = \int_a^b\ f(\gamma(t))^*\cdot f'(\gamma(t))\cdot \gamma'(t)\ dt = \int_a^b\ ((f\circ\gamma)(t))^*\cdot \frac{d}{dt}(f\circ\gamma)(t)\ dt = [\ |\ f(\gamma(t))\ |^2\ ]_a^b - \int_a^b\ \frac{d}{dt}((f\circ\gamma)^*)(t)...