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)...
My lecturer did the change of coordinates for a more general constant coefficient PDE \sum_{j=1}^n a_j\frac{\partial f}{\partial x_j}=b(x,u) in n-variables by defining the new variables as:
y_1=\frac{x_1}{a_1}
and
y_j=x_j-\frac{a_j}{a_1}x_1
How do you get this?
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...
Now I've got:
(G_1\frac{\partial r}{\partial x}+G_2\frac{\partial r}{\partial y}+G_3\frac{\partial r}{\partial z})-(G\frac{\partial r_1}{\partial x}+G\frac{\partial r_2}{\partial y}+G\frac{\partial r_3}{\partial z})+2G_1+2G_2+2G_3.
When I expand out the vectors(\frac{\partial r}{\partial x}...
0!
Which gives
((\nabla \times F).\nabla)r-(\nabla.r)(\nabla \times F)+2\nabla \times F
or
(G.\nabla)r-(\nabla.r)G+2G
One term less = a bunch of less components to deal with - I'll try expanding it out now and see where I get.
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...
So then the rectangle is an infinite set of discontinuous points as well, but we can integrate over it because we just use g(x,y) instead?
Why can't we do the same thing with a circle by changing to polar coordinates so that the domain can be defined like the rectangle in Cartesian? (I think...
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...
Wow, thanks a lot! Would never have thought of that - trig was one of my extra weaker points back in high school.
I've solved the non-homogenous case, gotten the resonance frequencies and now doing the homogenous case to get the general solution. The solution I've got for y''=\omega_0^2y=0...
Unfortunately the solution I got doesn't satisfy the original DE
Here's how they did it in the text:
2tv''-v'=-
Letting w=v'
2tw'-w=0
Separating variables and solving for w(t)
\therefore w(t)=v'(t)=ct^{\frac{1}{2}}
\therefore v(t)=\frac{2}{3}ct^{\frac{3}{2}}+k
So the annihilator...
Ah yeah, thanks for pointing that out. We haven't done Fourier series yet - I don't think it's even in the syllabus for this course but I'll have a look into it.
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...
The original equation is 2t2y''+3ty'-y=0
Let v be the unknown solution and y=vt-1. Differentiate y and y' and substitute it into the DE, work through the algebra and out pops 2tv''-v'=0.
Here're all the steps I took
2tv''-v'=0
v''-\frac{v'}{2t}=0
D(D-\frac{1}{2t})v=0
Let...
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...
Thanks, I think I get it - you mean the product rule, yeah?
(D+A)(D+B)y
=(D+A)(Dy+By)
= D2y+D(By)+ADy+ABy
= D2y+yB'+By'+ADy+ABy
= D2y+yB'+BDy+ADy+ABy
= (D2+B'+BD+AD+AB)y
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...
Letting d(x,y) be the diameter of the union of the sets, if x,y are in different sets, then d(x,y)≤d(x,z)+d(z,y)≤2M since every set contains z and the diameter of every set is bounded by M. Therefore d(x,y) is finite since 2M is finite.
If x,y are in the same set, then it would be bounded by...
This is what I'm thinking:
Let x,y,z be points in X. Given a metric d on X, we're to show the function d_1=log(1+d) is a metric on X.
<verify first 2 properties>
Consider (1+d(x,z))(1+d(z,y)). We have
<do working to show>
(1+d(x,z))(1+d(z,y)) \geq 1+d(x,y)
Taking the natural...
I have the diameter of the union of sets is less than or equal to the sum of the diameters of each set, and that's less than the sum of M (summed over each alpha). How might one go about working in the fact they all intersect?
(If two sets have a maximum diameter of M, and intersect, then I...
Thanks, this is what I have now:
(1+d(x,z))(1+d(z,y))=1+d(z,y)+d(x,z)+d(x,z)d(z,y)\geq 1+d(x,y)
Then taking logs of both sides
log((1+d(x,z))(1+d(z,y))) \geq log(1+d(x,y))
log(1+d(x,z))+log(1+d(z,y)) \geq log(1+d(x,y))