Wikipedia:Reference desk/Archives/Mathematics/2010 July 5

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July 5

Clifford Algebra Question

Is there a way to perform the map

$Ae_{01}+Be_{02}+Ce_{03}+De_{12}+Ee_{13}+Fe_{23}\mapsto GAe_{01}+HBe_{02}+ICe_{03}+JDe_{12}+KEe_{13}+LFe_{23}$

with "standard" functions acting upon the bivector on the left-hand side, where all the capital letters denote unknown, real-values constants? As it happens, the metric signature is $(4,0)$. By "standard functions", I mean all the derived Clifford products (wedge with some other element, commutator, exp etc.) acting upon the bivector on the left.--Leon (talk) 09:45, 5 July 2010 (UTC)

Integration

$\int {{\frac {1}{x{\sqrt {1-x}}}}dx}$?––Wikinv (talk) 23:24, 5 July 2010 (UTC)

Make the substitution $u={\sqrt {1-x}}$ and then use partial fraction decomposition.-Looking for Wisdom and Insight! (talk) 23:48, 5 July 2010 (UTC)

After doing the substitution, rather than immediately differentiating both sides, square both sides first so that from

$u^{2}=1-x\,$

$2u\,du=-dx\,$

And also, you get

$x=1-u^{2}\,$

Michael Hardy (talk) 17:58, 6 July 2010 (UTC)

Or let $x=\sin ^{2}u$. Depending on the set you're integrating over, that'd turn the integral into $\int {\frac {2\sin u\cos u}{\sin ^{2}u\cos u}}du$, which is on any integration table.—msh210℠ 18:24, 9 July 2010 (UTC)

Indeed. The integrals of 1/sin and 1/cos are the most difficult ones in first-year calculus, but they're in all the tables so no one has to do them from scratch. Michael Hardy (talk) 19:15, 10 July 2010 (UTC)