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)...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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|)...
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...
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...
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)...
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...
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...
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.
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...
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...
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
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...
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...
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...
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}...
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...
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...
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...
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...