Recent content by kuruman

Yes.

What happened to the cross term? Isn't it true that ##~\nabla^2(fg)=f\nabla^2g+2\mathbf{\nabla}f\cdot\mathbf{\nabla}g +g\nabla^2f~##?

You should draw the conclusion that you need to be careful when using the expression for the divergence in spherical coordinates lest there be point charge singularities at the origin. In this particular case you have an electric field in the radial direction $$E_r=Ae^{-\lambda r}(1+\lambda...

I would write two unit vectors $$\mathbf{\hat r}_i=\cos\phi_i\cos\lambda_i~\mathbf{\hat x}+\sin\phi_i\cos\lambda_i~\mathbf{\hat y}+\sin\lambda_i~\mathbf{\hat z}~~~~(i=1,2)$$ then take their dot product.

Let's take a few more steps back and apply the two basic conventions about ideal DC circuits: Batteries maintain a constant potential difference across their terminals; Straight line segments are equipotentials. Let's ground the negative terminal of the top battery to declare the potential at...

If I were observing volume changes of a plastic bottle, I would, measure the mass before and after by placing the bottle on a kitchen scale, just in case there is mass loss that would have to explain; place the bottle in a pot of water, fully immerse it, and put a mark with indelible ink at the...

OK, I understand that the ice melts because of the high pressure under the ice skate blade. How does that compare with the pressure inside a closed copper pipe full of water that bursts in extremely cold weather? Why isn't a new equilibrium established in this case so the water in the pipe...

I am wondering if someone better versed in non-equilibrium thermodynamics than me can attribute these, apparently initial-condition-dependent observations, to the Mpemba effect.

Did you weigh the bag every day to make sure that there was no mass loss through permeation of water vapor?

I think we can both agree on the following: 1. Using any number of methods, it can be shown that ##h=\sqrt{ab}.## 2. If either one of the segments, ##a## or ##b##, is defined as the unit segment, then ##h## represents the square root of the other segment in the chosen system of units.

It looks like that you lost track of what this construction is all about. It finds the square root of one of the segments, ##a## or ##b## that, together, make up the diameter. Note that ##\sqrt{1/2}=0.707## and ##\sqrt{2}=1.41.## Clearly, ##h=1## is the square root of neither ##a## nor ##b##...

I wouldn't put it quite that way. Starting with ##h^2=ab,## which we assume to be known, we can rewrite it in two ways: $$\begin{align} & \frac{h}{a}=\sqrt{\frac{b}{a}} \\ & \frac{h}{b}=\sqrt{\frac{a}{b}}. \end{align}$$ Simply put, it doesn't matter which of ##a## and ##b## you call the unit and...

The word philosophy literally means love of wisdom. In Plato's time, the distinction between a mathematician and a philosopher was not as sharp as it is today. The perceived connection of Plato to mathematics, which in pre-algebra days meant geometry, is illustrated by the traditional belief in...

Thanks, but how is that different from what we have been talking about? In Wallis's expression, let ##BP=h~;~~AB=b~;~~BC=a~## to obtain ##h=\sqrt{ab}~## which has been shown in many different ways here.

Here is a construction to clarify why units are important for the construction. Suppose you are given a line segment on a piece of paper (black ##b## in the figure on the right) and someone asked you to produce segment ##h## on the paper such that it is the square root of ##b##. You know from...