That would work too, but I'm not sure how I would go about building the rotation matrix from the principal axes -- I always learned rotations in terms of angles.
Hello,
I'm running a galaxy formation simulation. The output specifies the coordinates in (x, y, z) of all the particles in a galaxy, which usually fall in a disk. The orientation of the disk depends on the initial conditions, but it is generally not aligned with any of the coordinate axes...
Hi everyone. This isn't a specific homework problem, I'm just trying to understand a concept.
I've been studying the HI profiles (from the 21cm emission line). Almost all of them look like the attached profile. The velocity on the x-axis corresponds to the amount that the line has been...
Homework Statement
\lim_{x \to 0}[\frac{\sin(\tan(x))-\tan(\sin(x))}{x^7}]Homework Equations
\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} + ...
\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{215}+ ...The Attempt at a Solution
I have an idea of how to do this by replacing...
Homework Statement
For what values of p and q does \sum\limits_{n=2}^\infty \frac{1}{n^q\ln(n)^q} converge?
The Attempt at a Solution
I've tried a couple of tests, but given that there are two variables (p and q), I'm not really sure how to proceed. My hunch is that I have to use the...
It does seem pretty straightforward with integration by parts, but since I'm told to use the gamma function, I'd at least like to know how to do that.
If I use the substitution x=e^{-u/k}, I get dx = -k*e^{-u/k}\,du The integral then becomes \int^1_0 e^{-u}*-\frac{u}{k}*-k*e^{-u/k}\,du =...
Homework Statement
"Show that - \int^1_0 x^k\ln{x}\,dx = \frac{1}{(k+1)^2} ; k > -1.
Hint: rewrite as a gamma function.
Homework Equations
Well, I know that \Gamma \left( x \right) = \int\limits_0^\infty {t^{x - 1} e^{ - t} dt}.
The Attempt at a Solution
I've tried various substitutions...
I don't think so; it says to compute the derivatives of the function. Also, part b of the question asks me to use the Taylor expansion of cos(x) and compare it with the result from this part.
Homework Statement
"Determine the first two non-vanishing terms in the Taylor series of \frac{1-\cos(x)}{x^2} about x = 0 using the definition of the Taylor series (i.e. compute the derivatives of the function)."
So I know how compute the Taylor series about x=0; it involves finding f(0)...
A bit. The takeaways seem to be that BMR is difficult to calculate and varies quite a bit from person to person, and that a person might burn more calories if the trip takes longer. So I suppose that a trip that is mostly slow coasting might still burn a lot of energy.
What I'm more...
Do you use more energy cycling on, say, a 10 km flat stretch, or on a stretch where you spend the first 5km cycling uphill and the second 5km coasting downhill? What if you spend the first 1km cycling up a pretty steep incline, and the remaining 9km coasting down a more gentle slope? What if you...