By Campbell J.E.
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Extra info for A Course of Differential Geometry
992 *2 y , a We know 2 _ (a now Consider 23. Elliptic coordinates. * ,_ +u c 2 + '16 that the relations 2 -f u) (a 2 2 + v) (a 2 + w) __ (b 2 + u) 2 2 (b -f v) (b 4- w) ( ' ' give the coordinates of any point in space in terms of the and that the perpendiculars from focal coordinates u, v, w ; the centre on the threo confocals through any point are given 2 by _ - 2 ~~ ' = ~~ u)(vw) (v 2 4- 2 ~~ __ From p* = w) 2 (b 4- w) uw w 2 (a + u) cos 2 a + (6 2 + u) cos 2 + x cos a + y cos /? + z cos the tangent plane to the surface = and therefore 4 du, cfe a 2qdq = we now take w= = 2 g + 2 (c y u= dv, + p^ If + w) v the formula where is (c* + u) cos 2 y, (23.
These theorems, having served their purpose, disappear, as it were, from the calculus. There are some simple rules of the calculus which we now consider. The product two a tensor whose components are the products of each component of the first and each component of the second tensor. The upper integers of the of tensors is product are the upper integers of the two factors, and the lower integers of the product are the lower integers of the two Two factors. tensors of the same character that is, with the same number can be of each kind of integers, upper and lower which have the if we the take together added, components same we They can integers.
MNqT. 8) are tensor components. a very important theorem in the tensor calculus. It is the rule of taking what we call the tensor derivative TENSOR THEORY 14 and we see that the tensor derivative of a tensor component a tensor component. ;;:. P 11. Bules and definitions of the tensor calculus. We have now proved the most important theorem in the tensor calculus: its proof depended on the transformation theorems. These theorems, having served their purpose, disappear, as it were, from the calculus.
A Course of Differential Geometry by Campbell J.E.