Fast Fourier Analysis in action.
Any complex waveform, sound, or shape can be perfectly reconstructed as the sum of simple rotating circles (epicycles).
π can be reached through many elegant mathematical routes, including its decimal expansion, continued fractions, arccos(-1), arctangent combinations, definite integrals, infinite products, series like BBP and Ramanujan, and arithmetic-geometric mean iterations.
It is used to enable precise numerical computation of the constant in computational physics and cryptographic systems.
Whoahhh! This is one of the strangest things an earthquake can do. As the M7.8 earthquake hit the Philippines last week, the ground in this video appears to move almost like liquid.
👀 (Watch until the end)
This can happen because of liquefaction. It's when intense shaking causes water trapped in loose soil to build up pressure, making solid ground temporarily lose its strength and behave more like a fluid.
It’s why people can struggle to stand, and why areas near water with loose, saturated soil can see the ground ripple, sink or even “flow” during powerful earthquakes.
📹 Pangabuhi sa Uma Vlog
From areas to volumes and flows, integrals have specialized types.
The single variable integral ∫_a^b f(x) dx computes curve areas. Line integrals ∫_C f(x,y) ds sum along paths. Lebesgue integrals ∫_E f dμ generalize for complex functions. Double integrals ∬_R f(x,y) dx dy find surface volumes. Flux integrals ∬_S F · dS calculate surface flows. Triple integrals ∭_E f(x,y,z) dx dy dz determine volume contents.
They calculate volumes of custom 3D printed objects in manufacturing and analyze airflow in aircraft design.
Taylor series expand any smooth function around a point a using its derivatives at that point.
The formula is f(x) = Σ_{n=0}^∞ f^{(n)}(a)/n! (x-a)^n.
The graphic defines the components on the left and shows on the right how the linear (blue, n=1), quadratic (green, n=2), cubic (purple, n=3) and higher (orange) approximations approach the red curve y=f(x) near a. Color-coded terms below match the orders from constant to higher.
In real life, it is used to approximate functions in physics, engineering calculations, and computer algorithms for efficient evaluations.