MandysNotes

Lessons

Lessons

GaussianIntegral

 

 

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

Addition Formulae

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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?

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:

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

From the definition of cosine:

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

we find:

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

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:

\[ | \frac{z}{r} | = | \frac{z}{|z|} | = 1. \]

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:

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

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

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

NOTRad