Recent content by pasmith

You can use linear interpolation between the data points. Between z_i and z_{i+1} that gives you \int_{z_i}^z \frac{1}{\lambda(z)}\,dz = \int_{z_i}^z \frac{1}{A_i + B_iz}\,dz which you can do analytically.

Yes, where D is a Jordan normal form, ie. the expression of the map with respect to a basis of (generalized) eigenvectors \{v_1, \dots, v_n\}. Conjugation by P then gives the expression of the map with respect to the standard basis.

For the purpose of this exercise, does it make a difference to the mathematical analysis if the boundary condition is the dimensionally consistent kUt^p or U(t/t_0)^p rather than sloppy Ut^p? It is common to use scaled units in order sweep such constants of proportionality under the carpet. The...

Can you find a Liapunov function for the system?

I would assume the first, since the second would have been written as y \ln x. But the first leads to a non-linear, non-separable equation and the second leads to a linear equation.

A graphical description is not really a rigorous proof, although it might help you to find one. Ultimately, showing that X/\sim is homeomorphic to Y requires finding a continuous function f: X/\sim \to Y and showing that it has a continuous inverse. For example, showing that [0,1]/\sim where 0...

It should not be difficult to get from a graphical description to a parametrisation. For example, for the Mobius strip one can take a line segment with centre on a circle of radius R such that the angle to the horizontal plane goes through a half rotation as the centre moves through a full...

\epsilon < 8 renders as \epsilon < 8. Originally discovered in my post at https://www.physicsforums.com/threads/limit-of-piecewise-function-using-epsilon-delta.1081023/post-7267512. Looking at the preview of this post (with LaTeX not being rendered due to a known bug) suggests that < is being...

You haven't told us the definition of f. You can assume \epsilon < 8 (a \delta which works for such an \epsilon will also work for any larger \epsilon) so that 0 < 8 - \epsilon < 8 + \epsilon.

Most calculators will label it as "tan-1" rather than "arctan". In mathematics we generally measure angles in radians rather than degrees, so make sure that the calculator is set to the right unit.

How do you define [z] for z \in \mathbb{C}?

There is a fundamental problem here: \nabla \times (y + 5, 2z, y) = (-1,0,-1) is not zero, so (y + 5, 2z, y) cannot be the gradient of a function. y + 5 and y do not vanish simultaneously.

For some reason, the second displayed equation in the quote in this post is not being displayed. I can see when I edit the post that the LaTeX is there and is correct, but neither the LaTeX code nor the rendering thereof appears when looking at the post.

If the first equation is correct, then the second equation asserts that -\tan ka = \sqrt{\frac{1}{2\cos (2ka) + 2}} = \pm\frac 12 |\sec ka| which is clearly incorrect. Since you haven't shown your working for either derivation, we cannot see where you may have fallen into error.

I would start from \begin{split} \frac{\partial}{\partial \rho} &= \frac{\partial x}{\partial \rho}\frac{\partial}{\partial x} + \frac{\partial y}{\partial \rho}\frac{\partial}{\partial y} \\ &= \cos \phi \frac{\partial }{\partial x} + \sin \phi \frac{\partial}{\partial y} \\...