Recent content by penguino

Hence for every pi^ki there exists an ai in G such that pi^ki divides the order of ai. Somehow I must show that there is also such an ai for which the order has no prime factors other than pi, but I don't see how to do that.

Homework Statement (From an exercise-section in a chapter on Lagrange's theorem:) Let G be a finite abelian group and let m be the least common multiple of the order of its elements. Prove that G contains an element of order m. The Attempt at a Solution We have x^m = e for all x in G. By...

Yes, because \left(x^2+y^2\right)^3 = \left(r^2\cos^2\theta+r^2\sin^2\theta\right)^3 = r^2\left(\cos^2\theta+\sin^2\theta\right)^3 = \left(r^2\right)^3, and 4x^2y^2 = 4\left(r^2\cos^2\theta\right) \left(r^2\sin^2\theta\right) = r^4\left(4\cos^2\theta\sin^2\theta\right) = r^4\sin^2 2 \theta...

Sorry, I didn't mean to give you the impression that you should try this in cartesian coordinates. But can you show me how you did the conversion? Because I get: (x^2 + y^2)^3 = 4x^2y^2 Substituting x=r\cos \theta, y=r\sin \theta (r^2)^3=r^{4}4\sin^{2}\theta\cos^{2}\theta, r^2 = \sin^{2}2\theta.

Are you sure you did this conversion correct? Or are the loops described by \left(\sqrt{x^{2}+y^{2}}\right)^{3} = 4x^{2}y^{2}.? What's the integral for determining area in cartesian coordinates? What's the effect on an integral by substituting to polar coordinates? (What's the Jacobian...

What's the step you don't understand? \int \tan^{4}x \mathrm{d}x = \int \tan^{2}x \left(\sec^{2}x - 1\right) \mathrm{d}x or \int \tan^{2}x \left(\sec^{2}x - 1\right) \mathrm{d}x = \int \tan^{2}x \sec^{2}x \mathrm{d}x - \int \tan^{2}x\mathrm{d}x.