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The week that wasn't

Time is flying by fast and I can hardly comprehend where the week got spent. Atleast the day was spent well, redoing problems from a course on general relativity that I am currently sitting through. The concept of killing vectors is slowly sinking into me and I honestly can't get enough of the course, I wish he'd take classes everyday. It's surprising the amount of material the prof covers in a single hour and the last two weeks of classes have simply been mind blowing. Starting with the derivation of the equations of motion from the line element ds, it has been one blow after another.

On the other hand, I spent a bit of time thinking about simple integration problems that I could try the monte carlo method on and compare the errors that I would get with the errors from simpson's rule or trapezoid rule. One easy question that I could come up with was the volume of an n-dimensional sphere. It is fairly straight forward to convert a code that measures the area of a circle to the volume of a sphere in 3D, one just adds another random variable 'z' and extend the condition to '+z**2'. I've added the code at the end of this ipynb file. I haven't been able to figure out how to do the same integration using simpsons' or trapezoid rules but I shall. And I want to extend it to n-dimensions, that way I will be able to show how the error changes with number of points of integration for different dimensions.

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