Archive

"I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." Albert Einstein (Kollros 1956; Pais 1982, p. 216).

Now that we have a good dual basis for linear functionals on our tangent space, we can use it to express an important functional - the gradient of a function defined on our surface.

The tangent space and the metric lets us do basic calculus at a point on our surface: vector displacements with the metric telling us distances and angles.

The next step is to "do calculus" with scalar functions on our surface. At a point on our surface, we can approximate a scalar function with a Linear Functional These are what we are going to need:

At every point of our coordinate space, we can use the tangent space to define the distance in any (infinitesimal) direction. In general the square of the distance will be a quadratic form in the (infinitesimal) coordinate displacements along the basis vectors of the tangent space.

Let's set the scene of the things I think I understand, at least to the first approximation. We have a curvilinear coordinate system on a space under some metric. For example, the surface of a sphere. We will start simple, and then mess with it to look at a slightly more complicated variant.

Making diagrams for coordinate systems proved harder than I thought, and I got sidetracked.

So I've been getting back into a bit of physics, and decided to take a crack at general relativity and tensor calculus. It's all derivatives, right, so it should be easy? Well, of course, I'm having trouble. One major sticking point is the notion of a dual basis. I understand the definition but couldn't make it click into a substantive understanding. This is my attempt to start working it out.

Seems to be a rite of long passage.

We use Terraform a lot at my work. There's been an interesting learning curve as I've gone from no knowledge of it, to using it in production. I had a bit of trouble figuring out a conceptual model - a way of thinking about what it was doing - that worked for me.

Eventually I found one.

I worked a lot in my career on The Geometer's Sketchpad, an educational geometry visualization and interaction desktop application. Sketchpad had a long and useful life, finally retired as a product in 2019. Twenty-five years! It's still in use as an authoring environment for the HTML5 web viewer.

I'm not quite ready for a manifesto. Still gathering my own thoughts, and trying to understand software testing terminology and methodologies. But - feels like there is a fair amount of confusion about how it's supposed to work even in the best case.