The Gaussian integral I = ∫_{-∞}^∞ e^{-x²} dx is evaluated by squaring it to obtain the double integral ∬ e^{-(x²+y²)} dx dy.
In polar coordinates the Jacobian r and substitution u = r² reduce it to I² = π, hence I = √π. The derivation is accompanied by a 3D plot of
Mohr's Circle geometrically determines stresses on any inclined plane in plane stress conditions.
It plots σₙ on the horizontal axis and τₙ on the vertical, with its center at σ_avg = ½(σₓ + σᵧ). Original face stresses appear at A(σᵧ, τₓᵧ) and B(σₓ, −τₓᵧ). The circle crosses the σₙ axis at principal stresses σ₁ and σ₂, while peak shears reach ±½(σ₁ − σ₂). The pole allows orientation angles θ_p1 and θ_p2 to be read directly from lines to these points, matching the rotated stress elements displayed.
Mechanical engineers apply it to locate maximum shear planes in shaft design under torsion and bending to avoid material failure.
BITCOIN IS FALLING BEHIND GLOBAL M2.
Historically, BTC eventually catches up with liquidity.
This time, the reaction is simply taking longer.
THE GAP IS GETTING BIGGER.
THE PRESSURE IS BUILDING.
WHEN BITCOIN FINALLY MOVES, IT COULD BE EXPLOSIVE.
REMEMBER THIS POST.
The Gaussian integral I = ∫_{-∞}^∞ e^{-x²} dx is evaluated by squaring it to obtain the double integral ∬ e^{-(x²+y²)} dx dy.
In polar coordinates the Jacobian r and substitution u = r² reduce it to I² = π, hence I = √π. The derivation is accompanied by a 3D plot of the sy
Buoyant force is the upward force exerted by a fluid on an immersed object. According to Archimedes' principle, this force equals the weight of the fluid displaced by the object.
Fb = ρ × g × V
Where: ρ = density of the fluid
g = acceleration due to gravity
V = volume of fluid displaced
A denser fluid or a larger displaced volume produces a greater buoyant force. This simple principle explains why ships float and why objects feel lighter underwater.
The stock popped 19% on $1.596B in revenue, but AMC is still carrying $3.9B in debt, meaning the Gorilla lenders already collected before this quarter was even called a win.
What the Monkey holding shares is cheering is mostly what was left after the bill got paid. 🍌
BREAKING: $AMC jumped 19% in pre-market after posting its best quarterly revenue in company history.
Revenue hit $1.596 billion for the quarter, up 14.2% from last year.
Free cash flow came in at $190.1 million for the quarter.
CEO Adam Aron called it the best quarter in AMC's 106-year history.
Bilangan e ≈ 2,71828 merupakan dasar dari pertumbuhan kontinu dan salah satu konstanta paling penting dalam matematika.
• Pertumbuhan
e = lim(n→∞) (1 + 1/n)ⁿ
• Deret Tak hingga
e = Σ(n=0→∞) 1/n!
= 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...
• Logaritma
∫₁ᵉ (1/x) dx = 1
Tensor calculus is the language that lets mathematics work on curved surfaces, spacetime, and complex geometries.
• Tensor
T^(i₁...iₘ)_(j₁...jₙ)
A mathematical object that keeps its meaning even when coordinates change. It generalizes scalars and vectors.
• Metric Tensor
gᵢⱼ = eᵢ · eⱼ
Measures distances and angles in a space. It defines the geometry locally.
• Connection (Christoffel Symbol)
Γᵏᵢⱼ = ½ gᵏˡ (∂ᵢgⱼˡ + ∂ⱼgᵢˡ − ∂ˡgᵢⱼ)
Describes how vectors change as they move through curved space.
• Covariant Derivative
∇ₖTⁱⱼ = ∂ₖTⁱⱼ + ΓⁱₖˡTˡⱼ − ΓˡₖⱼTⁱˡ
Extends differentiation so it remains valid in curved coordinates and curved spaces.
• Einstein Field Equation
Gᵢⱼ = Rᵢⱼ − ½gᵢⱼR = (8πG/c⁴)Tᵢⱼ
Relates the curvature of spacetime to the distribution of matter and energy.
From measuring distances and angles to describing gravity itself, tensor calculus provides the mathematical framework behind differential geometry, general relativity, continuum mechanics, and much of modern physics.
If you have access, capital, relationships, or AI that gives you a 10x advantage and you don't use it, that's not integrity. That's stupidity.
https://t.co/eaHZpNRPMk…
Welcome to the Artemis Accords, Serbia! 🇷🇸
Serbia has joined 68 other nations by signing the Artemis Accords, and in doing so, has committed to the safe and peaceful exploration of space. https://t.co/2saHrV7FYq