Recent content by pasmith

Any CAS which knows that for ##a \in \mathbb{R}##, ##(a^2)^{1/2} = |a|## should be able to get it right.

Based on the sketch, it seems that ##\theta## is the angle of cylindrical polar coordinates, not spherical polar coordinates. I agree that the base of the cylinder is completely described by ##-\pi/2 \leq \theta \leq \pi/2##. The factor of 2 comes not from rotational symmetry about the origin...

It's a bit confusing to take ##b < a##; usually we have ##a < b##. My approach would be to define $$ g : [0,1]\to\mathbb{R} : x \mapsto f(b) + x(f(a) - f(b)) - f(b+ x(a - b)) \geq 0.$$ Now ##g(0) = g(1) = 0## and the constraint on the sign of ##g## allows you to say something about the signs of...

A Hessian matrix must be symmetric; this matrix is not. This is a consequence of the problem I noted in my earlier post: the vector field specified by the OP is not the gradient of a scalar function.

It may help to note that $$\frac{n^2}{n + \frac15} = n - \frac{1}{5 + \frac 1n}.$$

Yes. Expanding ##(z - a)^{-k}## in binomial series can be done in two ways; one converges for ##|z| < |a|## and the other for ##|z| > |a|##.

A lot o numerical analysis involves dealing with spaces of polynomials of degree at most $$N$$ on $$[-1,1]$$. There are various families of polynomials which can be taken as basis functions, and these will be orthogonal with respect to a particular inner product. Quite often, one wishes to...

The equation of a line in ##\mathbb{R}^2## can be written as $$\mathbf{n} \cdot (\mathbf{x} - \mathbf{x}_0) = 0$$ where ##\mathbf{x}_0## is a point on the line and ##\mathbf{n}## is a vector normal to the line. Here we have ##\mathbf{n} = (\cos \omega, \sin \omega)## and ##\mathbf{x}_0 = (5\cos...

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...