∮ Beautiful Equations
The Navier-Stokes Equation
Tha Navier-Stokes equation describes the motion of a parcel of fluid on Earth, either in the atmosphere or in liquid form (e.g. rivers, oceans).
Equation from Dr. Elie Bou-Zeid’s Environmental Fluid Mechanics class, Fall 2021.
D→uDt=∂→u∂t+→u⋅∇→u=−1ρ∇p+ν∇2→u+→g(1+ρ′ρ)−fc→k×→uWhere
- D→uDt is the material derivative of velocity with respect to time
- ρ is density
- p is pressure
- ν is viscosity
- g is gravity
- ρ′ is the difference between parcel density ρ and mean density ˉρ: (ρ′=ρ−ˉρ)
- fc is coriolis force: fc=2Ωsin(latitude)
- k is unit vector in vertical direction
The Bernoulli Equation
The Bernoulli equation derives from the Navier-Stokes equation and assumptions of (1) steady-state (∂→u∂t=0), (2) no viscous forces (ν=0), (3) no bouyant force (ρ=ˉρ⇒ρ′=0), and (4) no coriolois fc=0.
U222+p2ρ+gz2=U212+p1ρ+gz1Where
- Ui is total flow speed at point i (no flow normal to streamline)
- p is pressure at point i
- ρ is density
- g is acceleration due to gravity
- zi is height at point i relative to some baseline for all i
The Bernoulli equation is very useful for determining speed of flow due to pressure or elevation changes in pipes, rivers, etc.
Euler’s Formula
eiπ=cosx+isinxEuler’s Identity
A special case of Euler’s formula evaluated at x=π
eiπ+1=0Newton’s Basic Equation
This equation describes the motion, displacement, and acceleration of a body due to some force (or set of forces)
F=ma=m∂v∂t=m∂x2∂2tWhere
- F is force, in units N = kg m s−2
- m is mass, in units kg
- a is acceleration, in units m s−2
- v is velocity, in units m s−1
- x is displacement, in units m
- t is time, in units s
This equation can be used in all sorts of physical systems with various zeroeth, first, second-order, and higher-order forces driving the motion of a body
Fourier Sine and Cosine Series Representations of Functions
Any function on the interval 0≤x≤ℓ can be written as a sum of sines and cosines. Even functions can be written in terms of cosines only, and odd functions can be written in terms of sines only. Both sines and cosines must be present for a function with both even and odd components (general function).
Sine (odd functions)
f(x)=∞∑n=1ansin(nπxℓ)Where
an=2ℓ∫ℓ0f(x)sin(nπxℓ)dxCosine (even functions)
f(x)=∞∑n=0bncos(nπxℓ)Where
b0=1ℓ∫ℓ0f(x)dxand
bn=2ℓ∫ℓ0f(x)cos(nπxℓ)dx(n≥1)Fourier Series (any function)
f(x)=b0+∞∑n=1(ansin(nπx)⏟odd function+bncos(nπx)⏟even function)Fourier Transform and Inverse Fourier Transform
Let f be the function dependent on space or time (denoted by x) and let ˆf be the transformed function dependent on frequency (traditionally denoted by ζ). Then
ˆf(ζ)=∫∞−∞f(x)e−2πixζdx∀ζ∈Rand, inversely,
f(x)=∫∞−∞ˆf(ζ)e2πixζdζ∀x∈RThe Fourier transform changes a space-or-time dependent function into a frequency-dependent function. For example, if you plug an audio recording (time dependent) into a fourier transform, you would get a set of frequencies and their relative strengths in the signal back. The frequency function you get back can be inversely transformed into the original function; or, if you remove unwanted frequencies, you can inversely transform into an audio recording minus the unwanted high-pitched buzz.
Fourier transforms enable modern commincation and many forms of audio and visual data processing. Truly an amazing mathematical tool!