Recent content by LeonhardEuler

The Riemann Hypothesis is something else. It relates to the zeroes of the Riemann zeta function. This problem was solved in the 1600's.

The answer to that question is basically the fundamental theorem of calculus. To show enough to make this plausible, imagine you break the interval [a,b] into pieces. Call the pieces [a,x1], [x1,x2], [x2,x3] ... [xN-1,xN], [xN,b]. Assume they are equally large intervals and that xi+1-xi=h. Now...

Yes, it's interesting to look at other points of view, and I read those arguments sometimes too. It is surprising and counter-intuitive that complicated things like people and other animals could arise out of matter following physical laws. But it definitely doesn't violate the laws of...

Yes, atoms in a crystal can occasionally move past each other, but that is an extremely slow process, and it does not directly lead to a change in the overall macroscopic shape of an object. But in very old rocks the results of diffusion can be seen. The second law of thermodynamics doesn't...

True, but I think it's important to keep in mind the differences between a desk and a snow flake. For the sake of simplicity, assume the desk is made of iron, but there is no oxygen so it doesn't rust. Looking around I find the vapor pressure of water to be 600 Pa at 0 Celsius and the vapor...

Since a freezer takes in hotter air with a higher level of humidity than can remain at very low temperatures, more ice will tend to condense on the snowflake and it will lose its original shape fairly soon.

In addition to what everyone else said, it is not that much better of an explanation to say what parts of the brain are involved in some behavior than to just say the brain does it, in the sense that it adds little to the ability to make predictions about human behavior, or to understand how...

Don't forget that numbers like \pi and e are also irrational (among many others).

Also, here is another simple example of the same counter intuitive kind of thing. Suppose \int_{0}^{\infty}f(x,s)dx=1 for all s>0. You might think that \frac{\partial f}{\partial s}=0 But suppose f(x,s)=se^{-sx} Then \int_{0}^{\infty}se^{-sx}dx=1 but \frac{\partial f}{\partial s}=-s^2e^{-sx}+e^{-sx}

Really sorry about that. I meant K(x,s)=\sqrt{\frac{2}{\pi}}\frac{(b-a)e^{-\frac{(x-s)^2}{2\sigma^2}}}{\sigma[erf(\frac{s-a}{\sqrt{2}\sigma})+erf(\frac{b-s}{\sqrt{2}\sigma})]} And a and b are the limits of integration in the original problem, with the subscripts dropped because this is a 1D...

Yes, I am sure, and this is a question I get from a lot of people I have shown this to. Here is an example of a solution to the equation in one dimension...

Hello Everyone. An interesting equation has come in my thesis research, and I was wondering whether anyone had any useful information about it. It is this equation: \int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)K(x_1,...x_n,s_1...s_n)dx_1...dx_n=C K is a known function of the x's and s's. C is...

Yes. And the function x(t) is unknown. The solution to the problem is the entire function x(t), not some number. That is the difference between the calculus of variations and a regular minimization problem. The idea to take the derivative and set it to zero to minimize a function is that the...

That is right. What I meant was that the top of the cube alone is not a closed surface, it does not enclose the charge (or anything) by itself. Because of that, Gauss's law does not apply to the top surface alone. You apply Gauss's law to the cube as a whole, and by symmetry the flux out of...

Notice they ask about the flux through the top of the cube, not the whole cube. Gauss's Law applies to the flux through a closed surface enclosing a charge. Just the top of the cube is not a closed surface. Think about the symmetry of the problem, though.