Spherical coordinates: Adding a footnote for the cases of small n
Okumaya devam et...
← Previous revision | Revision as of 08:24, 4 May 2024 |
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| ==Spherical coordinates== | ==Spherical coordinates== |
| <!-- This section is linked from [[Spherical coordinate system]] --> | <!-- This section is linked from [[Spherical coordinate system]] --> |
| We may define a coordinate system in an {{math|''n''}}-dimensional Euclidean space which is analogous to the [[Spherical coordinates|spherical coordinate system]] defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate {{math|''r''}}, and {{math|''n'' − 1}} angular coordinates {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ..., ''φ''<sub>''n''−1</sub>}}, where the angles {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ..., ''φ''<sub>''n''−2</sub>}} range over {{math|[0, π]}} radians (or over {{math|[0, 180]}} degrees) and {{math|''φ''<sub>''n''−1</sub>}} ranges over {{math|[0, 2π)}} radians (or over {{math|[0, 360)}} degrees). If {{math|''x<sub>i</sub>''}} are the Cartesian coordinates, then we may compute {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}} from {{math|''r'', ''φ''<sub>1</sub>, ..., ''φ''<sub>''n''−1</sub>}} with:<ref>{{cite journal |last1=Blumenson |first1=L. E. |title=A Derivation of n-Dimensional Spherical Coordinates |journal=The American Mathematical Monthly |date=1960 |volume=67 |issue=1 |pages=63–66 |jstor=2308932 |doi=10.2307/2308932 }}</ref> | We may define a coordinate system in an {{math|''n''}}-dimensional Euclidean space which is analogous to the [[Spherical coordinates|spherical coordinate system]] defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate {{math|''r''}}, and {{math|''n'' − 1}} angular coordinates {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ..., ''φ''<sub>''n''−1</sub>}}, where the angles {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ..., ''φ''<sub>''n''−2</sub>}} range over {{math|[0, π]}} radians (or over {{math|[0, 180]}} degrees) and {{math|''φ''<sub>''n''−1</sub>}} ranges over {{math|[0, 2π)}} radians (or over {{math|[0, 360)}} degrees). If {{math|''x<sub>i</sub>''}} are the Cartesian coordinates, then we may compute {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}} from {{math|''r'', ''φ''<sub>1</sub>, ..., ''φ''<sub>''n''−1</sub>}} with:<ref>{{cite journal |last1=Blumenson |first1=L. E. |title=A Derivation of n-Dimensional Spherical Coordinates |journal=The American Mathematical Monthly |date=1960 |volume=67 |issue=1 |pages=63–66 |jstor=2308932 |doi=10.2307/2308932 }}</ref>{{efn|1=Formally, this formula is only correct for <math>n>3.</math> For <math>n-3,</math> the line beginning with <math>x_3=\cdots</math> must be omitted, and for <math>n=2,</math> the formula for [[polar coordinates]] must be used. The case <math>n=1</math> reduces to <math>x=r.</math> Using [[capital-pi notation]] and the usual convention for the [[empty product]], a formula valid for <math>n\ge 2</math> is given by <math display = inline>x_n =r\prod_{i=1}^{n-1}\sin \varphi_i</math> and <math display = inline>x_k =r \cos \varphi_k\prod_{i=1}^{k-1}\sin \varphi_i</math> for <math>k=1, \ldots, n-1.</math>}} |
| :<math>\begin{align} | :<math>\begin{align} |
Okumaya devam et...