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By Patrangenaru V.

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Explain the fallacy in the following argument: But j2 = -1. Therefore 1 = -1.

And so the rate of change of A with respect to 9. Now as But is trapped between , and so which is fixed, and must also tend to which tends to Therefore From this key result the area of the sector can be found by integration: area ? 7 in which (i) 5 dispositive (ii) r increases as 0 increases. Consider how the argument must be adapted if (a) be is negative (b) r decreases as 0 increases (c) both (a) and (b). Note that the final result remains the same in all cases. 5. SOLUTION The inner loop is formed as 9 varies from -to so its area is using cos2$ = 28 (1 + cos 20) Note Even though r is negative for the integrand 2 is always positive, so there is no problem of 'negative areas' as there is with curves below the x axis in cartesian co-ordinates.

39 de Moivre's theorem When you multiply two complex numbers (in polar form) you multiply their moduli and add their arguments: the product of Zi = ri(cos 0i + j sin 0J is and z2 = r2(cos 92 + j sin 0 2 ) ZiZ2 = rlr2(cos(9l + 0 2 ) + jsin(0i + 0 2 )). Much can be done by using this result repeatedly with just a single complex number z, of modulus 1. This allows you to concentrate on what happens to the argument. For if z = cos 0 + j sin 0 then z2 = cos (0 + 9) + j sin(0 + 9) = cos 20 + j sin 20, 3 z = 2 Z Z = cos (20 + 0) + j sin (20 + 0) = cos 30 + j sin 30, and so on.

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5 dimensional strictly locally homogeneous Riemannian manifolds by Patrangenaru V.

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