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

= 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
 * $$ x = 1 - u^2 \, $$
 * $$ x = 1 - u^2 \, $$

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

Or let $$x=\sin^2u$$. Depending on the set you're integrating over, that'd turn the integral into $$\int\frac{2\sin u\cos u}{\sin^2u\cos u}du$$, which is on any integration table.&mdash;msh210 &#x2120; 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)