Search results

  1. C

    Deriving spherical unit vectors in terms of cartesian unit vectors

    I'm trying to find the azimuthal angle unit vector \vec{\phi} in the cartesian basis by taking the cross product of the radial and \vec{z} unit vectors. \vec{z} \times \vec{r} = <0, 0, 1> \times <sin(\theta)cos(\phi), sin(\theta)sin(\phi), cos(\theta)> = <-sin(\theta)sin(\phi)...
  2. C

    Rotation of coordinates (context of solving simple PDE)

    If you rotate your rectangular coordinate system (x,y) so that the rotated x'-axis is parallel to a vector (a,b), in terms of the (x,y) why is it given by x'=ax+by y'=bx-ay I got x'=ay-bx, y'=by+ax from y=(b/a)x. By the way this is from solving the PDE aux+buy=0 by making one of the...
  3. C

    Prove this vector identity

    Homework Statement Show that: curl(r \times curlF)+(r.\nabla)curlF+2curlF=0, where r is a vector and F is a vector field. (Or letting G=curlF=\nabla \times F i.e. \nabla \times (r \times G) + (r.\nabla)G+2G=0) The Attempt at a Solution I used an identity to change it to reduce (?) it to...
  4. C

    Showing a metric space is complete

    Homework Statement Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)| is a complete metric space. The Attempt at a Solution Spent a few hours just thinking about this question, trying to prove...
  5. C

    Circle is a set of a discontinuities?

    Why is the characteristic function* of a ball in Rn continuous everywhere except on its surface?My lecturer said that a circle is a 'set of discontinuities' - what exactly does that mean? (some context: we're looking at how we can integrate over a ball. Previously we've only looked at Riemann...
  6. C

    2nd order ODE - undetermined coefficients?

    Homework Statement 1) Find the general solution of y''+ω02=Ccos3(ωx) 2) Show there exists two frequencies at which resonance occurs and determine them The Attempt at a Solution I've tried the method of undetermined coefficients, assuming a solution of the form y=(Acos(ωx)+Bsin(ωx))3...
  7. C

    Variable coefficient 2nd order DE

    Homework Statement This an example from Boyce+DiPrima's text on ODEs. Original problem is to solve 2t2y''+3ty'-y=0, given that y1=1/t is a solution. But I'm stuck at the part where I'm to solve 2tv''-v'=0, v being the second unknown solution. The Attempt at a Solution In the text...
  8. C

    Green's functions for ODEs, jump conditions

    Questions about Green's functions for ODEs, jump conditions I'm having a hard time understanding Green's functions which have been introduced quite early on in the course, and which I think hasn't been well motivated. I can't find any other resource which explains this at this level (have only...
  9. C

    Multiplying out differential operators

    In this video at around 9:00 , Carl Bender demonstrates a method of solving y''+a(x)y'+b(x)y=0. He first rewrites it in terms of differential operators D2+a(x)D+b(x))y(x)=0, then factors it (D+A(x))(D+B(x))y=0 then multiplies it out to determine B(x). I thought we would get...
  10. C

    Proof of equality of diameter of a set and its closure

    In showing diam(cl(A)) ≤ diam(A), (cl(A)=closure of A) one method of proof* involves letting x,y be points in cl(A) and saying that for any radius r>0, balls B(x,r) and B(y,r) exist such that the balls intersect with A. But if x,y is in cl(A), isn't there the possibility that x,y are...
  11. C

    Is the composition of a function and a metric a metric?

    Homework Statement Given that f is continuous and strictly increasing, f:[0, ∞)->[0, ∞), f(0)=0, and d(x,y) is the standard metric on the real number line, Is there a function f such that d'(x,y)=f(d(x,y)) is not a metric on the real number line? The Attempt at a Solution The standard...
  12. C

    Prove the diameter of a union of sets is finite

    Homework Statement Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A. The Attempt at a Solution The picture I have is if they all intersect (and assuming...
  13. C

    Show the following is a metric

    Homework Statement Let (X,d) is a metric space. Show that d_1=log(1+d) is a metric space. The Attempt at a Solution (it's not stated what d is so I'm assumed d=|x-y|) I've checked positivity and symmetry but am having trouble with showing the triangle inequality holds. i.e. log(1+|x-y|)...
  14. C

    How to measure the speed of light with this setup

    This is an experiment I'll be undertaking for labs. Given the following equipment: - laser with modulation input - lenses and mirrors - function generator - high-speed photo detectors - oscilloscope and the setup as shown in the picture (the function generator is connected to the...
  15. C

    Cartesian product of index family of sets

    Cartesian product of indexed family of sets The definition of a Cartesian product of an indexed family of sets (X_i)_{i\in I} is \Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\} So if I understand correctly, it's a function that maps every index i to an element f(i) such...
  16. C

    Construct a complex function with these properties

    Homework Statement Construct a function f:C \rightarrow C such that f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) (aside from the identity function) Hint: i^2=-1 what are the possible values of f(i). The Attempt at a Solution All I've been able to do so far is come up with some (hopefully correct)...
  17. C

    Prove that an upper bound a is the least upper bound

    Homework Statement Let A be a non-empty subset of R (real numbers) and a an upper bound in R for A. Suppose that every open interval I containing a intersects A (so the intersection is non-empty). Show that a is a least upper bound for A. The Attempt at a Solution I've seen the prettier...
  18. C

    Mathematica Solving ODE numerically in Mathematica - 'ndnum' error?

    I'm trying to solve this ODE R'(t)=\frac{-a}{R(t)^2} numerically in Mathematica (a, b are non-zero constants). Here's what I have: NDSolve[{R'[t]==-a/R[t]^2, R[0]==b, WhenEvent[R[t]==0, end=t; "StopIntegration"]}, R, {t,0,1}] It's returning with NDSolve:::ndnum : Encountered...
  19. C

    The union of a subset and its complement

    If S is a subset of X, and cS is the complement of S with respect to X, is the union of S and cS equal to X? Seems like a no-brainer but just want to be sure because I've yet to find a book that comments on this.
  20. C

    Current in an infinite current sheet

    Suppose we have a infinite current sheet of surface density \sigma and apply Ampere’s law to find the magnetic field. Using a rectangular loop of side lengths L, why would the enclosed current be I=\sigma L? Doesn't this imply the RHS has units of \frac{Q}{m}? Shouldn't we be looking at this...
  21. C

    Prove the W is the orthogonal complement of its orthogonal complement

    Homework Statement Let W be a subspace of R^n. Show that the orthogonal complement of the orthogonal complement of W is W. i.e. Show that (W^{\perp})^{\perp}=W The Attempt at a Solution This is one of those 'obvious' properties that probably has a really simple proof but which...
  22. C

    Proving the dor product of 4-vectors is Lorentz invariant

    Homework Statement Let A and B be 4-vectors. Show that the dot product of A and B is Lorentz invariant. The Attempt at a Solution Should I be trying to show that A.B=\gamma(A.B)? Thanks
  23. C

    Parameterize a union of circles

    Homework Statement Let C=\lbrace(x,y) \in R^2: x^2+y^2=1 \rbrace \cup \lbrace (x,y) \in R^2: (x-1)^2+y^2=1 \rbrace . Give a parameterization of the curve C. The Attempt at a Solution I'm not sure how valid it is but I tried to use a 'piecewise parameterisation', defining it to be...
  24. C

    Need help understanding proof that continuous functions are integrable

    Actually, the theorem is that functions that are uniformly continuous are Riemann integrable, but not enough room in the title! I'm failing to see the motivation behind proof given in my lecturer's notes (page 35, Theorem 3.29) and also do not understand the steps. 1) First thing I'm...
  25. C

    Gauss' law electrostatics problem involving charge densities

    Homework Statement A nonconducting spherical shell has a thickness b-a, where b is the outer radius and a the inner radius has a volume charge density \rho=\frac{A}{r}, r\in[a,b]. If there is a charge +q located at the center, what must A be in order for the electric field to be uniform in the...
  26. C

    Change of basis problem

    Edit complete, but it doesn't seem as though I can change the title. The latex arrows next to the 'P' aren't showing up for me but they're supposed to be left arrows Homework Statement Let B and C be bases of R^2. Find the change of basis matrices P_{B \leftarrow C} and P_{C\leftarrow B}...
  27. C

    Maximising light dispersion

    Homework Statement If light moves from a medium with a refractive index that is a function of wavelength, [\itex]n_1(\lambda)[/itex], to vacuum n_2=1, then dispersion will occur. Find the incident angle \theta_1 that will maximize dispersion. The Attempt at a Solution I'm interpreting...
  28. C

    Calculating errors and trendline - different correlation coefficients?

    Background: My lab group has taken a number of measurements of gas levels over 15 minute period every 15 seconds. We actually used a gas sensor and computer interface to do it so it pretty much did all the work. In calculating the errors for the data, using the uncertainties given in the user...
  29. C

    Elementary spring and mass question

    Homework Statement Two masses of equal mass m are attached by a single spring of sprint constant k, what is the resonant frequency of the system? The Attempt at a Solution I'm not sure how correct it is to treat the two masses on either end as single mass 2m. 2m\ddot{x}=-kx...
  30. C

    Determining stable and unstable equilibrium points

    Background: I've derived the equation of motion of the bead shown in the picture below using the Lagrangian \omega^2Rsin(\theta)+r\ddot{\theta}=0 The equilibrium points are at θ=0 and π. I'm now to show that only one of these points is stable. Homework Statement Show that only one of the...
Top