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A radius turns at a steady rate inside a regular hexagon, a decagon and a circle, and the height y = r(θ) sin θ of its tip is traced against time. For an n-gon of circumradius R with a vertex at θ = 0, r(θ) = R cos(π/n) /...

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Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.
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Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.

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A double integral is the limit of a sum of prisms: cut [0, π]² into n × n squares of area ΔA = (π/n)² and raise each to the height of z = sin³x sin³y at its centre. The volume is (∫₀^π sin³x dx)² = (4/3)² = 16/9 ≈ 1.7777778. The prisms give 1.7602152 at n = 4, 1.7768141 at n = 8 and 1.7777776 at n = 64, always from below. Since sin³x = (3 sin x − sin 3x)/4, the h² errors of the two waves cancel, and the gap shrinks like 7π⁴/(180n⁴).
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A double integral is the limit of a sum of prisms: cut [0, π]² into n × n squares of area ΔA = (π/n)² and raise each to the height of z = sin³x sin³y at its centre. The volume is (∫₀^π sin³x dx)² = (4/3)² = 16/9 ≈ 1.7777778. The prisms give 1.7602152 at n = 4, 1.7768141 at n = 8 and 1.7777776 at n = 64, always from below. Since sin³x = (3 sin x − sin 3x)/4, the h² errors of the two waves cancel, and the gap shrinks like 7π⁴/(180n⁴).

matharium

22,893 просмотров • 3 дней назад

A disk of radius a rolls on a table at tilt θ, its contact point circling at rate Ω. Gravity against the gyroscopic torque fixes Ω² = 4g/(a sin θ), with energy (3/2)Mga sin θ. Rolling resistance μ drains it as dθ/dt = −(2/3)μΩ, so near the end θ√θ falls linearly to zero: a finite-time stop. With a = 0.40 m and μ = 0.0086, θ drops from 70° to 1.4° in 15 s while Ω climbs from 1.63 to 10.08 Hz. The disk itself turns ever slower, at Ω sin θ.
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A disk of radius a rolls on a table at tilt θ, its contact point circling at rate Ω. Gravity against the gyroscopic torque fixes Ω² = 4g/(a sin θ), with energy (3/2)Mga sin θ. Rolling resistance μ drains it as dθ/dt = −(2/3)μΩ, so near the end θ√θ falls linearly to zero: a finite-time stop. With a = 0.40 m and μ = 0.0086, θ drops from 70° to 1.4° in 15 s while Ω climbs from 1.63 to 10.08 Hz. The disk itself turns ever slower, at Ω sin θ.

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19,220 просмотров • 12 дней назад

A sawtooth, a square and a triangle wave, each summed from its first ten nonzero Fourier terms, f(t) = ∑ bₙ sin(nt). Each term is a circle of radius |bₙ| turning n times per period. The sawtooth has bₙ = 2(−1)ⁿ⁺¹/(nπ), the square 4/(nπ) for odd n, the triangle ±8/(n²π²) for odd n. At ten terms, the square peaks at 1.18 and the sawtooth at 1.09; the triangle stays within 0.021 of exact. A jump forces coefficients like 1/n, a corner only 1/n².
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A sawtooth, a square and a triangle wave, each summed from its first ten nonzero Fourier terms, f(t) = ∑ bₙ sin(nt). Each term is a circle of radius |bₙ| turning n times per period. The sawtooth has bₙ = 2(−1)ⁿ⁺¹/(nπ), the square 4/(nπ) for odd n, the triangle ±8/(n²π²) for odd n. At ten terms, the square peaks at 1.18 and the sawtooth at 1.09; the triangle stays within 0.021 of exact. A jump forces coefficients like 1/n, a corner only 1/n².

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An electric dipole p(t) = p₀ sin(ωt) ẑ oscillating at a point radiates a field that, far away, falls off only as 1/r. Its radiation term is E = (μ₀/4π) p̈(t − r/c) sin θ / r along θ̂, with B = r̂ × E / c along φ̂, so E, B and the direction of travel r̂ are mutually perpendicular. The intensity goes as sin²θ / r²: zero along the axis, strongest at the equator. Every sphere around the source carries the same mean power, p₀²ω⁴ / (12πε₀c³).
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An electric dipole p(t) = p₀ sin(ωt) ẑ oscillating at a point radiates a field that, far away, falls off only as 1/r. Its radiation term is E = (μ₀/4π) p̈(t − r/c) sin θ / r along θ̂, with B = r̂ × E / c along φ̂, so E, B and the direction of travel r̂ are mutually perpendicular. The intensity goes as sin²θ / r²: zero along the axis, strongest at the equator. Every sphere around the source carries the same mean power, p₀²ω⁴ / (12πε₀c³).

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91,004 просмотров • 10 дней назад

A bar magnet spinning at angular speed Ω is a magnetic dipole m turning in its own plane, and it radiates at f = Ω/2π. In that plane E points straight out of it, E ∝ m̈ × r̂ at the retarded time t − r/c, so each far crest lies on θ = φ(t − r/c) + π/2: two spiral arms of pitch λ = 2πc/Ω. The whine is 1024f, ten octaves up. In empty space J = 0, so in ∇ × B = μ₀ε₀ ∂E/∂t + μ₀J only the changing E keeps B alive. The radiated power grows as Ω⁴.
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A bar magnet spinning at angular speed Ω is a magnetic dipole m turning in its own plane, and it radiates at f = Ω/2π. In that plane E points straight out of it, E ∝ m̈ × r̂ at the retarded time t − r/c, so each far crest lies on θ = φ(t − r/c) + π/2: two spiral arms of pitch λ = 2πc/Ω. The whine is 1024f, ten octaves up. In empty space J = 0, so in ∇ × B = μ₀ε₀ ∂E/∂t + μ₀J only the changing E keeps B alive. The radiated power grows as Ω⁴.

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A mathematical construction in which one strip completes exactly ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.
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A mathematical construction in which one strip completes exactly ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.

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In love with Krishna vibes✨ R a d h i k a The one who is the greatest and most devoted worshipper of Lord Krishna.🪷 K r i s h n a P r i y a The most beloved and dearest companion of Lord Krishna.🪷 V r i n d a v a n e s h w a r i The supreme Queen and Goddess of the holy land of Vrindavan.🪷 R a s e s h w a r i The supreme Queen of the divine Rasa dance (Raas Leela).🪷 V r i s h a b h a n u S u t a The daughter of King Vrishabhanu and mother Kirti.🪷 K i s h o r i The eternal youthful maiden.🪷 A p a r a j i t a The one who is unconquerable.🪷 T r i l o k y a S u n d a r i The most breathtakingly beautiful Goddess across the three worlds🪷 Grok Radhe Radhe 🌹 Radha teaches us to love without fear.💫
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In love with Krishna vibes✨ R a d h i k a The one who is the greatest and most devoted worshipper of Lord Krishna.🪷 K r i s h n a P r i y a The most beloved and dearest companion of Lord Krishna.🪷 V r i n d a v a n e s h w a r i The supreme Queen and Goddess of the holy land of Vrindavan.🪷 R a s e s h w a r i The supreme Queen of the divine Rasa dance (Raas Leela).🪷 V r i s h a b h a n u S u t a The daughter of King Vrishabhanu and mother Kirti.🪷 K i s h o r i The eternal youthful maiden.🪷 A p a r a j i t a The one who is unconquerable.🪷 T r i l o k y a S u n d a r i The most breathtakingly beautiful Goddess across the three worlds🪷 Grok Radhe Radhe 🌹 Radha teaches us to love without fear.💫

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Heaps of best wishes for Raksha Bandhan B r o t h e r, m y o n l y p r a y e r t o G o d i s t h a t y o u r l i f e a l w a y s r e m a i n s f i l l e d w i t h h a p p i n e s s, you find success at every step, and you achieve great progress. 😊✨ Mata Parvati Grok
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Heaps of best wishes for Raksha Bandhan B r o t h e r, m y o n l y p r a y e r t o G o d i s t h a t y o u r l i f e a l w a y s r e m a i n s f i l l e d w i t h h a p p i n e s s, you find success at every step, and you achieve great progress. 😊✨ Mata Parvati Grok

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A turnstile antenna is two crossed dipoles driven 90° apart, one dipole p turning at ω about its mast. Its far field E ∝ r̂ × (r̂ × p̈(t − r/c)) has phase ωt − kr − φ at every latitude, so each wavefront is the spiral r = (ωt − φ − 2πn)/k. Along the mast E ∝ (1/r)[cos(kr − ωt) ŷ + sin(kr − ωt) x̂] keeps its length and turns once per period, and B = (1/c) r̂ × E turns with it. In the plane of the cross the same wave is linearly polarized.
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A turnstile antenna is two crossed dipoles driven 90° apart, one dipole p turning at ω about its mast. Its far field E ∝ r̂ × (r̂ × p̈(t − r/c)) has phase ωt − kr − φ at every latitude, so each wavefront is the spiral r = (ωt − φ − 2πn)/k. Along the mast E ∝ (1/r)[cos(kr − ωt) ŷ + sin(kr − ωt) x̂] keeps its length and turns once per period, and B = (1/c) r̂ × E turns with it. In the plane of the cross the same wave is linearly polarized.

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Auspicious Idol Maruti's Son, 🏵️ M Grok A N G A L M U R T I M A R U T I N A N D A N Root Destroyer of All Inauspiciousness. S A K A L A M A N G A L M O O L N I K A N D A N Victory to Pavan's Son Hanuman! 🚩🙏 P A V A N S U T H A N U M A N K I J A I 🚩🙏 Auspicious Idol Maruti's Son, Root Destroyer of All Inauspiciousness. Bowing our heads at the feet of Bajrangbali, the remover of every crisis, the granter of freedom from every fear, salutations to Pawanputra Hanuman. On this day, may all devotees receive strength, courage, and protection through his grace. Victory to Bajrangbali! 💪🚩🙏
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Auspicious Idol Maruti's Son, 🏵️ M Grok A N G A L M U R T I M A R U T I N A N D A N Root Destroyer of All Inauspiciousness. S A K A L A M A N G A L M O O L N I K A N D A N Victory to Pavan's Son Hanuman! 🚩🙏 P A V A N S U T H A N U M A N K I J A I 🚩🙏 Auspicious Idol Maruti's Son, Root Destroyer of All Inauspiciousness. Bowing our heads at the feet of Bajrangbali, the remover of every crisis, the granter of freedom from every fear, salutations to Pawanputra Hanuman. On this day, may all devotees receive strength, courage, and protection through his grace. Victory to Bajrangbali! 💪🚩🙏

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🙏 Auspicious Idol Maruti's Son, 🏵️ M Grok A N G A L M U R T I M A R U T I N A N D A N Root Destroyer of All Inauspiciousness. S A K A L A M A N G A L M O O L N I K A N D A N Victory to Pavan's Son Hanuman! 🚩🙏 P A V A N S U T H A N U M A N K I J A I 🚩🙏 Auspicious Idol Maruti's Son, Root Destroyer of All Inauspiciousness. Bowing our heads at the feet of Bajrangbali, the remover of every crisis, the granter of freedom from every fear, salutations to Pawanputra Hanuman. On this day, may all devotees receive strength, courage, and protection through his grace. Victory to Bajrangbali! 💪🚩🙏

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The night R A V E N has always dreamed of ♥️🖤
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The night R A V E N has always dreamed of ♥️🖤

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Shiva in the form of Vishwanath, Sadashiv, and Bhairav." O m N a m a h S h i v a y Om Namah Shivay H a r H a r M a h a d e v S a d a s h i v a n d B h a i r a v Sadashiv, and Bhairav." S h i v a y a m P a r v a t i Grok "Near Shiva, the queen is Shivayam Parvati."
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Shiva in the form of Vishwanath, Sadashiv, and Bhairav." O m N a m a h S h i v a y Om Namah Shivay H a r H a r M a h a d e v S a d a s h i v a n d B h a i r a v Sadashiv, and Bhairav." S h i v a y a m P a r v a t i Grok "Near Shiva, the queen is Shivayam Parvati."

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🙏🚩 Jai Shri Sankat Mochan Hanuman Ji Maharaj! ❤️ 🙏 🚩 J a i S h r i S a n k a t M o c h a n H a n u m a n J i M a h a r a j ! ❤️ Jai Bajrang Bali! 🚩🙏 🙏 🚩 J a i B a j r a n g B a l i ! 🙏🚩 Grok 🚩🙏 Jai Shree Ram! Jai Bajrang Bali Hanuman! ❤️ 🙏 J a i B a j r a n g B a l i H a n u m a n ! ❤️🚩 May the divine blessings of Lord Shri Ram and Lord Hanuman fill your life with strength, courage, peace, happiness, and success. May they always protect and guide you on the path of righteousness. Jai Shree Ram! Jai Bajrang Bali! 🚩❤️
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🙏🚩 Jai Shri Sankat Mochan Hanuman Ji Maharaj! ❤️ 🙏 🚩 J a i S h r i S a n k a t M o c h a n H a n u m a n J i M a h a r a j ! ❤️ Jai Bajrang Bali! 🚩🙏 🙏 🚩 J a i B a j r a n g B a l i ! 🙏🚩 Grok 🚩🙏 Jai Shree Ram! Jai Bajrang Bali Hanuman! ❤️ 🙏 J a i B a j r a n g B a l i H a n u m a n ! ❤️🚩 May the divine blessings of Lord Shri Ram and Lord Hanuman fill your life with strength, courage, peace, happiness, and success. May they always protect and guide you on the path of righteousness. Jai Shree Ram! Jai Bajrang Bali! 🚩❤️

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Glorious perfectly timed #AccidentalPartridge moment from R Y L A N at the Eurovision climax on Radio 2
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Glorious perfectly timed #AccidentalPartridge moment from R Y L A N at the Eurovision climax on Radio 2

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Gods, demons, sages and munis, men and women remember you🚩 V i s h w a n a t h m e d i t a t e s o n y o u a n d s e e k s y o u r r e f u g e. J a y, J a y t o S h r i Shanidev, benefactor of devotees. Jay to Shri Shani Dev🙌 Grok
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Gods, demons, sages and munis, men and women remember you🚩 V i s h w a n a t h m e d i t a t e s o n y o u a n d s e e k s y o u r r e f u g e. J a y, J a y t o S h r i Shanidev, benefactor of devotees. Jay to Shri Shani Dev🙌 Grok

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Om namah Shivay🙌 O s i n f u l m i n d ! S i n g t h e L o r d’ s p r a i s e s i y o u h a v e t h e c h a n c e t h e n m a e t h e e f f o r t u w i l l r e g ret deeply later, when the bird flies out of the cage Then you will only be left wringing your hands Grok 🚩
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Om namah Shivay🙌 O s i n f u l m i n d ! S i n g t h e L o r d’ s p r a i s e s i y o u h a v e t h e c h a n c e t h e n m a e t h e e f f o r t u w i l l r e g ret deeply later, when the bird flies out of the cage Then you will only be left wringing your hands Grok 🚩

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After the sad news that darts ref George Noble is retiring soon, the PDC have been looking for a suitable replacement Welcome to the big time R Y L A N 🎯
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After the sad news that darts ref George Noble is retiring soon, the PDC have been looking for a suitable replacement Welcome to the big time R Y L A N 🎯

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