Ok, I solved it. For potential future persons with a similar problem: the matrix elements (in this case the elastic constants and compliance coefficients) are symmetric about the diagonal, which means that c12=c21, c13=c31 etc. I wrongly assumed that all elements below the diagonal were zero...
Homework Statement
(I'm trying to replicate some results in an academic paper where they have calculated elastic properties of a crystal. Because I'm going to do a lot of similar time-consuming calculations following this one, I need to learn how to do them using a computer.)
The compliance...
Homework Statement
[/B]
Calculate \widehat{Y^{2}}
(i.e., the mean of the square of Y.
Homework Equations
Y=\sum_{k=0}^{N-1}y_{k}
where
y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k}
and
t=N\tau
The quantities y_{k} (or G_{k}) are statistically independent.
The Attempt at a Solution...
Thanks. That gives me
\sum_{k=0}^{N-1}e^{\gamma \tau k}\int_{k\tau}^{(k+1)\tau}F'(t')dt'.
What's left now is to move the exponential into the integrand, but I'm not sure how that can be justified.
Homework Statement
Show that
\sum_{k=0}^{N-1}e^{\gamma \tau k}\int_{0}^{\tau}F'(k\tau+s)ds
can be written as
\int_{0}^{t}e^{\gamma t'}F'(t')dt'
Homework Equations
1. t=N\tau
2. \int_{0}^{\tau}F'(k\tau+s)ds has the same statistical properties for each interval of length \tau, and is...
I'm not self-taught and have, now that I think about it, indeed used the chain rule before. It is, however, definitely one of the rules of differentiation that I've had the least practice on.
Ok, so I checked your work and get
\frac{dv}{d\tau}=-N\gamma v_0 e^{-\gamma\tau...
Just confusion on my part. It's my textbook that claims the solution to the differential equation can be written in that form.
Ok, so differentiating w.r.t. \tau the chain rule gives me, first of all, that
\frac{d\int_{0}^{\tau}F'(k\tau+s)ds}{dt} =\frac{1}{N}(k+1)F'(\tau(k+1)).
Is this...
Homework Statement
This is actually a problem from my physics textbook, but I think it's mostly a mathematical problem, which is why I post it here:
Show that the Langevin equation
1: \frac{dv}{dt}=-\gamma v+\frac{1}{m} F'(t)
is solved by
2...
Homework Statement
I've been given that the Bessel function
∫(J3/2(x)/x2)dx=1/2π (the integral goes from 0 to infinity).
Homework Equations
∫(J3/2(ax)/x2)dx,
where a is a constant.
The Attempt at a Solution
Is the following correct?
a2∫(J3/2(ax)/(ax)2)dx=a2/2π
(This...
I'd really need some help.
Substituting ∆\tau in the last expression for
\frac{\delta p ∆v\tau}{l}
(a rearrangement of the Clausius-Clapeyron equation, and using the fact that δT and ∆\tau are equal), I get
v=\frac{\kappa bc\delta p \tau}{2mgal}(\frac{1}{\rho_i}-\frac{1}{\rho_w})...
No one?
Perhaps I should state my answer to b) explicitly: the speed with which the bar sinks is
v=\frac{∆z}{∆t}=\kappa\frac{∆\tau}{2mga}.
Is this correct?
Homework Statement
Question no. 4 in this document (there's a helpful picture, too):
Homework Equations
The Clausius-Clapeyron equation:
\frac{\delta p}{\delta \tau}=\frac{l}{\tau ∆v},
where v is the volume per unit mass, i.e., the inverse of the density.
The Attempt at a...