Manuel Cardoso @ManuelCard68882 - Twitter Profile

6 months ago

the fourier transform is how you see structure inside signals time domain lies to you frequency domain exposes it any signal → sum of pure oscillations sines and cosines are the basis what it does: → decomposes a function into frequencies → tells you “what frequencies exist” and “how strong they are” continuous form: F(ω) = ∫ f(t) e^(−iωt) dt discrete world: → DFT / FFT → turns sampled data into usable spectra why it matters: in engineering: → filter noise → design communication systems → analyze vibrations, audio, RF in physics: → waves, heat, quantum states → solutions become algebra in frequency space in computing: → image compression (JPEG) → signal processing pipelines → fast convolution via frequency multiplication key idea: → convolution in time = multiplication in frequency → complexity collapses interpretation: → sharp spikes → dominant frequencies → spread spectrum → complex / noisy signal the fourier transform is not about math it’s about changing perspective to make systems tractable

oprydai's tweet photo. the fourier transform is how you see structure inside signals

time domain lies to you
frequency domain exposes it

any signal → sum of pure oscillations
sines and cosines are the basis

what it does:
→ decomposes a function into frequencies
→ tells you “what frequencies exist” and “how strong they are”

continuous form: F(ω) = ∫ f(t) e^(−iωt) dt

discrete world:
→ DFT / FFT
→ turns sampled data into usable spectra

why it matters:

in engineering:
→ filter noise
→ design communication systems
→ analyze vibrations, audio, RF

in physics:
→ waves, heat, quantum states
→ solutions become algebra in frequency space

in computing:
→ image compression (JPEG)
→ signal processing pipelines
→ fast convolution via frequency multiplication

key idea:
→ convolution in time = multiplication in frequency
→ complexity collapses

interpretation:
→ sharp spikes → dominant frequencies
→ spread spectrum → complex / noisy signal

the fourier transform is not about math
it’s about changing perspective to make systems tractable

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@oprydai

5 months ago

maxwell’s equations define how fields behave, not just forces they replace “action at a distance” with local interactions in space and time four constraints: → gauss (electric) ∇·E = ρ/ε₀ charge density creates divergence in the electric field → gauss (magnetic) ∇·B = 0 no magnetic sources, only loops → faraday ∇×E = −∂B/∂t changing magnetic field induces circulation in E → ampere – maxwell ∇×B = μ₀J + μ₀ε₀ ∂E/∂t currents and changing E generate B structure: → divergence ties fields to sources → curl ties fields to dynamics key move: → maxwell added the displacement current term (∂E/∂t) without it, symmetry breaks and waves don’t exist result: → combine curls → wave equations → fields sustain each other and propagate c = 1 / √(μ₀ε₀) light = electromagnetic wave implications: → no medium required → information carried by field oscillations in engineering: → antennas radiate by accelerating charges → circuits leak energy as radiation at high frequency in computation: → full-wave solvers approximate these PDEs interpretation: → not four separate laws → a coupled system defining allowable field evolution maxwell’s equations are a constraint system on reality everything electromagnetic must satisfy them