Project: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

$P=\left\{[x_{0},x_{1}),[x_{1},x_{2}),\dots ,[x_{n-1},x_{n}]\right\},$

Default ${\textstyle E=\Sigma mc^{2}}$

Default $E=\Sigma mc^{2}$

$$E=\Sigma mc^{2}$$

be a partition of I, where

$a=x_{0}<x_{1}<x_{2}<\cdots <x_{n}=b$$a^{2}{\frac {2}{3}}{3}$

The Riemann sum of f over I with partition P is defined as

$S=\sum _{i=1}^{n}f(x_{i}^{*})(x_{i}-x_{i-1}),\quad x_{i-1}\leq x_{i}^{*}\leq x_{i}.$

The choice of $x_{i}^{*}$ in the interval $[x_{i-1},x_{i}]$ is arbitrary.

Example: Specific choices of $x_{i}^{*}$ give us different types of Riemann sums:

• If $x_{i}^{*}=x_{i-1}$ for all i, then S is called a left Riemann sum.
• If $x_{i}^{*}=x_{i}$ for all i, then S is called a right Riemann sum.
• If $x_{i}^{*}={\tfrac {1}{2}}(x_{i}+x_{i-1})$ 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

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$S=\sum _{i=1}^{n}v_{i}(x_{i}-x_{i-1}),$

where $v_{i}$ is the supremum of f over $[x_{i-1},x_{i}]$, then S is defined to be an upper Riemann sum.

• Similarly, if $v_{i}$ is the infimum of f over $[x_{i-1},x_{i}]$, then S is a lower Riemann sum.

The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:

$r={\sqrt {x^{2}+y^{2}}}\quad$ (as in the Pythagorean theorem or the Euclidean norm), and

$\varphi =\operatorname {atan2} (y,x)\quad$,

where atan2 is a common variation on the arctangent function defined as

$\operatorname {atan2} (y,x)={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}$

References