trigonometry-is-my-bitch
fouriestseries:

Taylor Series Approximations
A Taylor series is a way to represent a function in terms of polynomials. Since polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.
There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].
The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.
Mathematica code:
f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]

fouriestseries:

Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomialsSince polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.

There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].

The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.

Mathematica code:

f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]
fuckyeahfluiddynamics

fuckyeahfluiddynamics:

Ferrofluidscolloidal suspensions made up of ferromagnetic nanoparticles and a carrier liquid—are known for their interesting and sometimes bizarre behaviors due to magnetic fields. The video above shows how, when subjected to an increasing magnetic field, a single droplet of a ferrofluid on a superhydrophobic surface will split into several droplets. The process is called static self-assembly, and it results from the ferrofluid seeking a minimum energy state relative to the force supplied by the magnetic field. Change the magnetic field and the droplets shift to the next energy minimum. But what happens when you change the magnetic field continuously and too quickly for the droplets to respond? A whole different set of structures and behaviors are observed (video link). This is dynamic self-assembly, a different ordered state only achieved when the ferrofluid is forceably kept away from the energy minima seen in the first video. For more, see the additional videos and the original paper. (Video credit: J. Timonen et al.; via io9)