Tensors for Beginners

Tensors for BeginnersAlbert TarantolaSeptember 15, 20041 Tensor NotationsThe velocity of the wind at the top of Eiffel s tower, at a given moment, can be representedby avectorvwith components, in some local, given, basis,{vi}(i=1, 2, 3). The velocityof the wind is defined at any pointxof the atmosphere at any timet: we have avector fieldvi(x,t).The water s temperature at some point in the ocean, at a given moment, can be repre-sented by ascalar T. The fieldT(x,t)is ascalar state of stress at a given point of the Earth s crust, at a given moment, is representedby asecond order tensor with components{ ij}(i=1, 2, 3;j=1, 2, 3).

Obvious examples of differentiable manifolds are the lines and surfaces of ordinary ge-ometry. Our 3-D physical space (with, possibly, curvature and torsion) is also represented by a differentiable manifold. The space-time of general relativity is a four dimensional dif-ferentiable manifold. Acoordinatesystemmaynot“cover”allthemanifold.

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  G eometry, Ometry

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