# MandysNotes

### Lessons

Lessons
Tuesday, 13 November 2012 19:33

### A Quick and Dirty Derivation of the Maxwell Distribution

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Suppose we are given a certain quantity of an ideal gas at some fixed temperature, and we want to know what sort of distribution of velocities to associate with this gas.
That is, given a range of velocities, $\Delta v = v_\beta - v_\alpha,$
what is the number of molecules, $\Delta n,$with velocities in the region of phase space $\Delta v= \Delta v_x \Delta v_y \Delta v_z?$

Tuesday, 13 November 2012 19:19

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${\color{Goldenrod}\sin}\ \left( {\color{MidnightBlue}z_{1}} + {\color{MidnightBlue}z_{2}} \right) = {\color{Goldenrod}\sin}\ {\color{MidnightBlue}z_{1}} \ {\color{Blue}\cos}\ {\color{MidnightBlue}z_{2}} \ + \ {\color{Blue}\cos}\ {\color{MidnightBlue}z_{1}}\ {\color{Goldenrod}\sin}\ {\color{MidnightBlue}z_{2}} \$

${\color{Blue}\cos}\ \left( {\color{MidnightBlue}z_{1}} + {\color{MidnightBlue}z_{2}} \right) = {\color{Blue}\cos}\ {\color{MidnightBlue}z_{1}} \ {\color{Blue}\cos}\ {\color{MidnightBlue}z_{2}} \ - \ {\color{Goldenrod}\sin}\ {\color{MidnightBlue}z_{1}}\ {\color{Goldenrod}\sin}\ {\color{MidnightBlue}z_{2}} \$

Tuesday, 13 November 2012 19:13

### Half-Angle Formulae

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${ {\color{Goldenrod}\sin}\ \frac{{\color{MidnightBlue}z} }{2} } = \pm \left( \frac{1- {\color{Blue}\cos}\ {\color{MidnightBlue}z} }{2} \right)^{\frac{1}{2} }$

${ {\color{Blue}\cos}\ \frac{{\color{MidnightBlue}z} }{2} } = \pm \left( \frac{1+ {\color{Blue}\cos}\ {\color{MidnightBlue}z} }{2} \right)^{\frac{1}{2} }$

Tuesday, 13 November 2012 18:53

### Derivatives of Trigonometric Functions

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$\frac{d}{dx}{ \color{Goldenrod}\sin}\ {z} = {\color{Blue}\cos}\ {z}$

$\frac{d}{dx}{\color{Blue} \cos}\ {z} = -{\color{Goldenrod} \sin}\ {z}$

Tuesday, 13 November 2012 18:03

### Primitives of Trigonometric Functions

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$\int{ \color{Goldenrod}\sin}\ {z} = - {\color{Blue}\cos}\ {z}$

$\int{\color{Blue} \cos}\ {z} ={\color{Goldenrod} \sin}\ {z}$

Sunday, 12 February 2012 22:57

### Standard Deviations and Cuttlefish

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Question:

What's the difference between sample standard deviation and population standard deviation? When do we use N-1 and when N in the denominator?

### Half Angle Formulae

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We can find the half angle formula for cosine by starting with the double angle formula:

$\cos{(2\alpha)} = \cos^{2}{\alpha} - \sin^{2}{\alpha} \ .$

Making the substitution:

$\gamma = 2\alpha ,$

leads to the equation:

$\cos{\gamma} = \cos^{2}{(\frac{\gamma}{2})} - \sin^{2}{(\frac{\gamma}{2})} .$

### Addition and Double Angle Formulae

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From the definition of cosine:

$\cos{\theta} =\frac{e^{i\theta} + e^{-i\theta}}{2}$

we find:

$2\cos{(\alpha + \beta)} = e^{i(\alpha + \beta)} + e^{-i(\alpha + \beta)} .$

### The Pythagorean Theorem from Euler’s Formula

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Consider any complex number $z, \$with modulus

$\sqrt{z\bar{z} } = |z| = r \ ,$

where $r \in \mathbb{R^{+}}\ .$

Note that $\frac{z}{r} \$would be a complex number with modulus one:

$| \frac{z}{r} | = | \frac{z}{|z|} | = 1.$

### Sum of Squares Using Euler's Formula

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Euler's formula makes life much easier when it comes to deriving trigonometric identities.

Note, first of all, that $|e^{i\alpha}| = 1,$gives us:

$\left( \text{Re} (e^{i\alpha}) + i\text{Im}(e^{i\alpha}) \right) \left( \text{Re}(e^{i\alpha}) – i\text{Im}(e^{i\alpha}) \right)$

$= \left( \cos{\alpha} + i\sin{\alpha} \right) \left( \cos{\alpha} - i\sin{\alpha} \right)$

$= \cos^{2}{\alpha} + \sin^{2}{\alpha} = 1.$

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