Matrices and Determinants Questions

Question Number 100068 by  M±th+et+s last updated on 24/Jun/20

$${hello}\:{all}\:{i}\:{have}\:{some}\:{questions}? \\$$$$\\$$$$\left.\mathrm{1}\right){what}\:{is}\:{the}\:{riemann}\:{hypothesis}? \\$$$$\\$$$$\left.\mathrm{2}\right){how}\:{did}\:{they}\:{determine}\:{the}\:{distance} \\$$$${to}\:{the}\:{sun}? \\$$$$\\$$$$\left.\mathrm{3}\right){how}\:{did}\:{we}\:{measure}\:{the}\:{speed}\:{of}\:{light}? \\$$$$\\$$

Answered by JDamian last updated on 24/Jun/20

$${You}\:{can}\:{find}\:{the}\:{answers}\:{in}\:{google} \\$$

Commented by  M±th+et+s last updated on 24/Jun/20

$${thank}\:{you}\:{for}\:{your}\:{comment}\:{sir}. \\$$$${but} \\$$$${in}\:{google}\:{there}\:{is}\:{no}\:{mathmatics}\: \\$$$${explaning}\:{just}\:{a}\:{simple}\:{explanation}. \\$$

Answered by smridha last updated on 25/Jun/20

$$\:\boldsymbol{{M}}{a}\boldsymbol{{xwell}}\:\boldsymbol{{diff}}:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{said}}\:\boldsymbol{{us}} \\$$$$\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}=\boldsymbol{\mu}_{\mathrm{0}} \epsilon_{\mathrm{0}} \frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}}{\partial{t}^{\mathrm{2}} }\:\boldsymbol{{and}}\:\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}=\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} \frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}}{\partial{t}^{\mathrm{2}} } \\$$$$\boldsymbol{{where}}\:\overset{\rightarrow} {\boldsymbol{{E}}}=\boldsymbol{{electric}}\:\boldsymbol{{field}} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\rightarrow} {\boldsymbol{{B}}}=\boldsymbol{{magnetic}}\:\boldsymbol{{field}} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mu}_{\mathrm{0}} =\boldsymbol{{magnetic}}\:\boldsymbol{{permiability}}\:\boldsymbol{{in}}\:\boldsymbol{{free}}\:\boldsymbol{{space}} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\epsilon}_{\mathrm{0}} =\boldsymbol{{permitivity}}\:\boldsymbol{{of}}\:\boldsymbol{{free}}\:\boldsymbol{{space}} \\$$$$\left[\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{besically}}\:\mathrm{3}\boldsymbol{{d}}\:\boldsymbol{{wave}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \right. \\$$$$\boldsymbol{{this}}\:\boldsymbol{{two}}\:\boldsymbol{{eq}}^{\boldsymbol{{ns}}} \:\boldsymbol{{describe}}\:\boldsymbol{{light}}\:\boldsymbol{{as}} \\$$$$\left.\boldsymbol{{a}}\:\boldsymbol{{electromagnetic}}\:\boldsymbol{{wave}}\right] \\$$$$\:\:\boldsymbol{{now}}\:\boldsymbol{{analysis}}\:\boldsymbol{{the}}\:\boldsymbol{{dimention}}\:\boldsymbol{{of}} \\$$$$\boldsymbol{{the}}\:\boldsymbol{{product}}\:\boldsymbol{{of}}\:\mu_{\mathrm{0}} \:\boldsymbol{{and}}\:\boldsymbol{\epsilon}_{\mathrm{0}} .. \\$$$$\:\:\:\boldsymbol{{dim}}\left[\boldsymbol{\epsilon}_{\mathrm{0}} \boldsymbol{\mu}_{\mathrm{0}} \right]=\left[\boldsymbol{{M}}^{−\mathrm{1}} \boldsymbol{{L}}^{−\mathrm{1}} \boldsymbol{{T}}^{\mathrm{4}} \right].\left[\boldsymbol{{ML}}^{−\mathrm{1}} \boldsymbol{{T}}^{−\mathrm{2}} \right] \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[\boldsymbol{{L}}^{−\mathrm{2}} \boldsymbol{{T}}^{\mathrm{2}} \right] \\$$$$\boldsymbol{{now}}\:\boldsymbol{{dim}}\left[\frac{\mathrm{1}}{\sqrt{\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} }}\right]=\left[\boldsymbol{{L}}.\boldsymbol{{T}}^{−\mathrm{1}} \right]=\boldsymbol{{dim}}\left(\boldsymbol{{v}}\right) \\$$$$\boldsymbol{{where}}\:\boldsymbol{{v}}=\boldsymbol{{velocity}}\:\boldsymbol{{but}}\:\boldsymbol{{this}}\:\boldsymbol{{not}} \\$$$$\boldsymbol{{any}}\:\boldsymbol{{odinary}}\:\boldsymbol{{velocity}}\:\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{very}} \\$$$$\boldsymbol{{special}},\:\boldsymbol{{the}}\:\boldsymbol{{speed}}\:\boldsymbol{{of}}\:\boldsymbol{{light}}\:\boldsymbol{{in}} \\$$$$\boldsymbol{{free}}\:\boldsymbol{{space}}\:\boldsymbol{{denoted}}\:\boldsymbol{{by}}\:\boldsymbol{{C}}.. \\$$$$\boldsymbol{{so}}\:\boldsymbol{{our}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{looks}}\:\boldsymbol{{like}}: \\$$$$\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}=\frac{\mathrm{1}}{\boldsymbol{{C}}^{\mathrm{2}} }\frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}}{\partial\boldsymbol{{t}}^{\mathrm{2}} }\:{and}\:\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}=\frac{\mathrm{1}}{\boldsymbol{{C}}^{\mathrm{2}} }.\frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}}{\partial\boldsymbol{{t}}^{\mathrm{2}} } \\$$$$\boldsymbol{{now}}\:\boldsymbol{{speed}}\:\boldsymbol{{of}}\:\boldsymbol{{light}}: \\$$$$\:\:\boldsymbol{{C}}=\frac{\mathrm{1}}{\sqrt{\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} }}=\frac{\mathrm{1}}{\sqrt{\mathrm{4}\boldsymbol{\pi}×\mathrm{10}^{−\mathrm{7}} ×\mathrm{8}.\mathrm{854}×\mathrm{10}^{−\mathrm{12}} }} \\$$$$\:\:\:\:\:=\mathrm{2}.\mathrm{99795638}×\mathrm{10}^{\mathrm{8}} \approx\mathrm{3}×\mathrm{10}^{\mathrm{8}} \boldsymbol{{m}}.\boldsymbol{{s}}^{−\mathrm{1}} \\$$$$\boldsymbol{{here}}\:\boldsymbol{{I}}\:\boldsymbol{{used}}\:\boldsymbol{{the}}\:\boldsymbol{{experimental}}\:\boldsymbol{{value}} \\$$$$\boldsymbol{{of}}\:\boldsymbol{\mu}_{\mathrm{0}} \:\boldsymbol{{and}}\:\boldsymbol{\epsilon}_{\mathrm{0}} . \\$$

Commented by Rasheed.Sindhi last updated on 25/Jun/20

$$\mathcal{G}\overset{\frown} {\mathcal{O}}\overset{\frown} {\mathcal{O}D}\:{use}\:{of}\:{mathematical}\:{notation} \\$$$${in}\:{the}\:\boldsymbol{{field}}\:{of}\:\boldsymbol{{human}}\:\boldsymbol{{emotions}}! \\$$

Commented by  M±th+et+s last updated on 24/Jun/20

$${very}\:{nice}\:{explanation}\:. \\$$$${god}\:{bless}\:{you} \\$$

Commented by smridha last updated on 24/Jun/20

$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\boldsymbol{{thank}}\right)_{\boldsymbol{{i}}} \\$$

Commented by Rasheed.Sindhi last updated on 25/Jun/20

$${Can}\:{we}\:{consider}\: \\$$$$\boldsymbol{{human}}-\boldsymbol{{emotions}}-\boldsymbol{{field}} \\$$$${as}\:{a}\:\boldsymbol{{mathematical}}-\boldsymbol{{field}}? \\$$$${Of}\:{course},\:{why}\:{not} \\$$$$\boldsymbol{{if}}\:\:{we}\:{can}\:\boldsymbol{{define}}\:\boldsymbol{{thanks}}, \\$$$$\boldsymbol{{sorry}}\:\&\:{other}\:{human}-{emotions}! \\$$

Commented by smridha last updated on 25/Jun/20

$$\boldsymbol{{yeah}}\:\boldsymbol{{why}}\:\boldsymbol{{not}}!!\boldsymbol{{after}}\:\boldsymbol{{all}}\: \\$$$$\boldsymbol{{M}}{a}\boldsymbol{{thematics}}\:\boldsymbol{{is}}\:\boldsymbol{{nothing}}\:\boldsymbol{{more}}\:\boldsymbol{{than}}\: \\$$$$\boldsymbol{{a}}\:\boldsymbol{{language}}.\boldsymbol{{if}}\:\boldsymbol{{there}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{wish}} \\$$$$\boldsymbol{{there}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{way}}.. \\$$