By B Lautrup

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10 Show that for arbitrary vectors a, b, c, and d. 75) i δi j δ j k = δik j Copyright © 2005 IOP Publishing Ltd. 13 Show that the Levi-Civita symbol is completely antisymmetric in all three indices, i j k = − ikj = − j ik = − kj i . 83) i j k i j k = 6. 86) ∇ × (∇ × v) = ∇(∇ · v) − (∇ · ∇)v. 87) Where in these relations does it make sense to remove the parentheses? 55) transforms as a vector, ∇i = rotation. 17 Show that the trace j ai j ∇ j under an arbitrary i Tii of a tensor is invariant under a rotation.

C .. A gravitational field line cannot form a closed path C. × ...... .... . . . P .... . .. .. x .... . . .... .. . 0 .. P . . × A closed curve C = P − P can be viewed as the difference between two paths connecting the same points. 40 3. GRAVITY √ − , where is the gravitational potential in the satellite’s orbit. Calculate the velocity of a satellite moving at ground level. 3 Arthur C. Clarke proposed (Wireless World, October 1945, pp 305–308) that communication satellites should be put into a circular equatorial orbit with revolution time equal to Earth’s rotation period, so that the satellites would stay fixed over a point at the equator.

Field lines around a point particle all come in from infinity and converge upon the particle. 34 Ö 3. GRAVITY ÇÃ x−x origin to an arbitrary point x , the collective gravitational field due to all the material particles in a volume V becomes an integral, ÖÖ x ................................... ......... ....... ....... ...... .. ... . .. .. ... .... .... . ... .... ....