matharium (@matharium) / X

A plane in three dimensions is fixed completely by one point on it and a normal vector. n · (r - r₀) = 0 Because r - r₀ always lies in the plane, its dot product with n is zero. Expanding gives the Cartesian form ax + by + cz = d, while r = r₀ + su + tv sweeps the plane

matharium

@matharium

9h

Monte Carlo integration estimates an integral by averaging the integrand at random points. ∫Ω f(x) dx ≈ V(Ω) · (1/N) Σᵢ₌₁ᴺ f(xᵢ) Error ∝ 1/√N whatever the dimension, so grid methods whose cost grows like a power of d lose in high dimensions. In one or two dimensions it

matharium

@matharium

11h

An ECG looks nothing like a sine wave. It is still made entirely of them. Fourier's theorem says any repeating signal is a sum of sines, so a heartbeat has to be one too. One sine gives a hum. Five produce a lump where the spike belongs. Twenty sharpen the QRS complex and reveal

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matharium

@matharium

13h

Can you reverse the spin of these snakes?

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Two products of vectors: one returns a number, the other returns a vector. a · b = ‖a‖ ‖b‖ cos θ and a × b = ‖a‖ ‖b‖ sin θ n In coordinates the dot product is a₁b₁ + a₂b₂ + a₃b₃, while the cross product is the determinant with rows i, j, k, then a₁ a₂ a₃, then b₁ b₂

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