By John Earman

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85) where now ∂ (x, y, z) ∂x ∂ y ∂z ∂ y ∂z ∂x ∂z ∂x ∂ y = + + ∂ (u, v, w) ∂u ∂v ∂w ∂u ∂v ∂w ∂u ∂v ∂w ∂z ∂ y ∂x ∂x ∂z ∂ y ∂ y ∂x ∂z − − − . 86) 28 1 Mathematical Tools • Problem 3 - Line, Surface, and Volume Elements: (a) The square of the length of a short path, ds, in cartesian coordinates is given by ds 2 = d x 2 + dy 2 + dz 2 . What is ds expressed in polar, cylindrical, and spherical coordinates? (b) Transform the cartesian surface element d xd y to polar coordinates. (c) Calculate the cartesian volume element d xd ydz in cylindrical and spherical coordinates.

114) z¯ = a − ib is called the complex conjugate of z = a + ib. Instead of the notation z¯ some authors use z ∗ . In particular the sum z + z¯ is always real, whereas the difference z − z¯ is always imaginary. 5 Complex Numbers† 37 Fig. 11 Representing the complex number z by a vector imaginary axis z=a+ib ib i 1 a real axis Some rules: • adding complex numbers: (a + ib) + (c + id) = (a + c) + i (b + d) • subtracting complex numbers: (a + ib) − (c + id) = (a − c) + i (b − d) • multiplying complex numbers: (a + ib) (c + id) = ac + iad + ibc + i 2 bd = (ac − bd) + i (ad + bc) • dividing complex numbers: a + ib ac + bd bc − ad (a + ib) (c − id) = = 2 +i 2 2 c + id c + d c + d2 + id) − id) (c (c Notice that the first two operations, unlike the next two, do not mix real and imaginary parts.

The situation is one of static equilibrium. We assume however that Sisyphus pushed just hard enough to overcome the force of gravity by a ‘negligible’ amount without causing ‘noticeable acceleration’. 36 1 Mathematical Tools Wpath II = dr · F + horizontal dr · F vertical =0 because d r ⊥ F= Fg = d r · Fg = −s Fg sin α . vertical Again friction is neglected. We observe that in our special example both paths lead to the same result. Mathematically we express this via d r · F(r ) − d r · F(r ) = path I d r · F(r ) = 0 .