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...