View source for Testwiki:VisualEditor testing/TestMath

Let ''f'' : ''D'' → '''R''' be a function defined on a subset, ''D'', of the real line, '''R'''. Let ''I'' = [''a'', ''b''] be a closed interval contained in ''D'', and let 
:<math>P= \left \{[x_0,x_1),[x_1,x_2),\dots,[x_{n-1},x_{n}] \right \},</math> 
:Default <math display="inline">E=\Sigma mc^2</math>  
:Default <math>E=\Sigma mc^2</math> <math display="block">E = \Sigma mc^2</math> 
be a partition of ''I'', where 
:<math>a=x_0<x_1<x_2<\cdots<x_n=b</math><math>a^2{\frac {2} {3}}{3}</math>
The '''Riemann sum''' of ''f'' over ''I'' with partition ''P'' is defined as 
:<math>S = \sum_{i=1}^{n} f(x_i^*)(x_{i}-x_{i-1}), \quad x_{i-1}\le x_i^* \le x_i.</math> 
The choice of <math>x_i^*</math> in the interval <math>[x_{i-1},x_i]</math> is arbitrary. 

'''Example:''' Specific choices of <math>x_i^*</math> give us different types of Riemann sums:
* If <math>x_i^*=x_{i-1}</math> for all ''i'', then ''S'' is called a '''left Riemann sum'''.
* If <math>x_i^*=x_i</math> for all ''i'', then ''S'' is called a '''right Riemann sum'''.
* If <math>x_i^*=\tfrac{1}{2}(x_i+x_{i-1})</math> for all ''i'', then ''S'' is called a '''middle Riemann sum'''.
* The average of the left and right Riemann sum is the '''trapezoidal sum'''.
* If it is given that
* 
:wow oweo 
::<math>S = \sum_{i=1}^{n} v_i(x_{i}-x_{i-1}),</math> 
:where <math>v_i</math> is the [[supremum]] of ''f'' over <math>[x_{i-1},x_i]</math>, then ''S'' is defined to be an '''upper Riemann sum'''. 
* Similarly, if <math>v_i</math> is the [[infimum]] of ''f'' over <math>[x_{i-1},x_i]</math>, then ''S'' is a '''lower Riemann sum'''.

The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r''&nbsp;≥&nbsp;0 and ''φ'' in the interval (−π, π] by:<ref>{{Cite book|first=Bruce Follett|last=Torrence|coauthors=Eve Torrence|title=The Student's Introduction to Mathematica|year=1999|publisher=Cambridge University Press|isbn=0-521-59461-8}}</ref>

:<math>r = \sqrt{x^2 + y^2} \quad</math> (as in the [[Pythagorean theorem]] or the [[Euclidean norm]]), and
:<math>\varphi = \operatorname{atan2}(y, x) \quad</math>,
where [[atan2]] is a common variation on the [[arctangent]] function defined as
:<math>\operatorname{atan2}(y, x) =
\begin{cases}
\arctan(\frac{y}{x}) & \mbox{if } x > 0\\
\arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
\arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
-\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
\text{undefined} & \mbox{if } x = 0 \mbox{ and } y = 0
\end{cases}</math>
References<references />