Community articles — Math

The density of solid water, unlike most molecules, is less than that of its liquid form. Its precise value is of use in many applications. Freezing a spherical droplet of water and analyzing the changed shape from a sphere to a sphere with a slight peak in order to find the density of ice. We find the density of ice to be at 0.90 ± 1.66 · 106 g/mL. The precision of our measurement was limited by uncertainty in the angle measurements of the peak of the droplet.

A proof is given that the summation of all prime numbers can be assigned the value of 13/12, as well as values that can be assigned to the summation of all multiples and all odd multiples.

When measuring a speed, the most common way to calculate it is by recording how far something went and the time it took to go that far. In the case of light, this is very difficult. One could conceivably shine a light over a vast distance and have someone else record when they see the light, but this would be difficult even at large distances. The person recording when they see it will need to have terrific reflexes to accurately measure a correct time as the time will be very short. A better method involves the use of a quickly rotating mirror and a beam of light. By aiming a beam of light o the rotating mirror, then reflecting it o a second stationary mirror back into the rotating mirror, calculations can be made on the speed of light. After first hitting the rotating mirror, the mirror will rotate very slightly in the time it takes the beam of light to return and will reflect back to a different position from where it came from. By measuring the displacement of the round trip, a measurement of the speed of light can be made.

Göteborg universitet, Chalmers universitet

The differential wave equation can be used to describe electromagnetic waves in a vacuum. In the one dimensional case, this takes the form $\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 0$. A general function $f(x,t) = x \pm ct$ will propagate with speed c. To represent the properties of electromagnetic waves, however, the function $\phi(x,t) = \phi _0 sin(kx-\omega t)$ must be used. This gives the Electric and Magnetic field equations to be $E (z,t) = \hat{x} E _0 sin(kz-\omega t)$ and $B (z,t) = \hat{y} B _0 sin(kz-\omega t)$. Using this solution as well as Maxwell's equations the relation $\frac{E_0}{B_0} = c$ can be derived. In addition, the average rate of energy transfer can be found to be $\bar{S} = \frac{E_0 ^2}{2 c \mu _0} \hat{z}$ using the poynting vector of the fields.

Preinforme guia de laboratorio L4.

Résolution de problème inverse par point fixe

Deze ding

A very quick and easy to understand introduction to Gram-Schmidt Orthogonalization (Orthonormalization) and how to obtain QR decomposition of a matrix using it.