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

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July 5

Clifford Algebra Question

Is there a way to perform the map

$A{e}_{01}+B{e}_{02}+C{e}_{03}+D{e}_{12}+E{e}_{13}+F{e}_{23}\mapsto GA{e}_{01}+HB{e}_{02}+IC{e}_{03}+JD{e}_{12}+KE{e}_{13}+LF{e}_{23}$

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

$u^2=1-x$

$2udu=-dx$

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)