Copasetic Flow

This installment in the ‘It’s Obvious. Not!’ series relates to the second edition of the book “div grad curl and all that” by H.M. Schey, published by W. W. Norton. Near the end of the example I referenced here, the author of “div grad curl and all that” states that the following integral is ‘trivial’ and results in an answer of 1/6 pi, (specifically, this falls on page 26 of the second edition). As far as I can tell, the solution is more tedious than it is trivial. I’m hoping there really is a trivial solution. If you know it, please add it to the comments below. I’m posting two versions of the ‘tedious’ solution here. The integral in question: The author suggests switching to polar coordinates before solving the integral using the following substitutions: The substitution that’s not mentioned is: So, now to solve the ‘trivial’ integral, first use the substitutions mentioned above: Factoring out the -r squared term in square root: Using the trigonometry...

This starts a series of posts that hopefully will add detail if not clarity to a number of physics and math topics. As I have been reviewing several of my physics and math texts, I’ve noticed that there are a number of ‘missing steps’ in some of the explanations and examples. In some cases, they seem to have been overlooked, and in others, the author of the book has actually made that statement dreaded by students everywhere, ‘the derivation is trivial/obvious…’ Each of these posts will take one of these ‘incomplete’ texts and elucidate the, (usually brief), steps that I found useful in understanding them. If you have other clarifications that are helpful, please add them! This first post relates to an excellent summary of vector calculus titled: “div grand curl and all that: an informal text on vector calculus”. In the second edition of the book on p. 25 the author sets out to take the partial derivatives of: He immediately arrives at the conclusion that: My confusion:...