Homework Statement
Let B be an m×n matrix with complex entries. Then by B* we denote the n×m matrix that is obtained by forming the transpose of B followed by taking the complex conjugate of each entry. For an n × n matrix A with complex entries, prove that if u*Au = 0 for all n × 1 column...
[PLAIN]http://img815.imageshack.us/img815/9894/fourier5313.png [Broken]
there was a minus sign. oops.
In any case, relevant equations:
F_{c}(f(x,t)) = \sqrt{\frac{2}{\pi}} \int^{\infty}_{0} cos (\lambda x) f(x,t) dx
F_{c}(f''(x,t)) = -\sqrt{\frac{2}{\pi}} f'(0,t) - \lambda^{2} F_{c}(f(x,t))...
Homework Statement
Find the solution u, via the Fourier sine/cosine transform, given:
u_{tt}-c^{2}u_{xx}=0
IC: u(x,0) = u_{t}(x,0)=0
BC: u(x,t) bounded as x\rightarrow \infty , u_{x}(0,t) = g(t)
2. The attempt at a solution
Taking the Fourier transform of the PDE, IC and BC...
Well, then I would have
m_{2}\ddot{x_{2}}-m_{1}\ddot{x_{1}} = -kq
...which, from what I see, doesn't do a whole lot because I still can't factor the left side to do anything.
Homework Statement
Two masses, m1 and m2, are connected to each other by a spring with a spring constant k. The system moves freely on a horizontal frictionless plane. Find the natural frequency of oscillation.
Homework Equations
F = -kx
F = ma
The Attempt at a Solution
Let m1 be the mass...
Homework Statement
Given u_tt = F(x,t,u,u_x, u_xx), give the finite difference approximation of the pde (ie using u_x = (u(x + dx; t) - u(x - dx; t))/(2dx) etc.)
Homework Equations
Well, clearly, u_x = (u(x + dx; t) - u(x - dx; t))/(2dx)
The Attempt at a Solution
I really have no idea how...
Homework Statement
u_t = -{{u_{x}}_{x}}
u(x,0) = e^{-x^2}
Homework Equations
The Attempt at a Solution
The initial state is a bell curve centred at x=0. The second partial derivative of u at t=0 is {4x^2}{e^{-x^2}}, which is a Gaussian function, which means nothing to me other than its...
tiny-tim, thanks for those. desperately needed.
hallsofivy, I did draw the diagram, and found that it was rather inconveniently symmetrical, which was why I got stumped. going by your suggestion, from θ = 0 to θ = π/2, I would be integrating along the right side of the curve, where the...
Homework Statement
Find
\int{\int_{D}x dA}
where D is the region in Q1 between the circles x2+y2=4 and x2+y2=2x using only polar coordinates.
The Attempt at a Solution
Well, the two circles give me r=2 and r=2 cos \theta, and the integrand is going to be r2cos \theta, but I have no...
*backtracks*
ooooooooooookay. so.
when x=0, f_x = -2, but f_x = 0, so x=0 is not a solution.
which brings me back to the points (-1,0) and (1,2). looking at my calculations for (-1,0), turns out that it's horrendously wrong.
At (-1,0), f_xx = -2(6-2) = -8, D = (-8)(-2) - (0-3+2)^2 = 16 - 1...
well, from f_y, I can deduce that x=0 or yx^2 - x - 1 = 0.
If x=0, from f_x, y= -2.
If yx^2 - x - 1 = 0,
y = (x+1)/(x^2)
Subbing into f_x,
-2(x^2 - 1)(2x) - 2((x+1)-x-1)(2x(x+1)/(x^2) - 1) = 0
-2(x^2 - 1)(2x) = 0
x(x^2 - 1) = 0
x = 0 (as above), x = 1 or x = -1
If x = 1, y = 2
If x = -1...