Tensor CalculusDate: 03/06/98 at 18:19:53 From: Ben Saucer Subject: Tensors and General Relativity Dear Dr. Math: I have never been able to understand tensor calculus. I would like a "layman's" explanation of the Ricci tensor, the Weyl tensor, and the geometric interpretation of the components of each. Why are there ten components of each? What is the "metric" of space? I am familiar with scalar and vector calculus, "four-vectors," and "Minkowskian" space. Date: 03/06/98 at 19:11:11 From: Doctor Anthony Subject: Re: Tensors and General Relativity Tensors are a further extension of ideas we already use when defining quantities like scalars and vectors. A scalar is a tensor of rank zero, and a vector is a tensor of rank one. You can get tensors of rank 2, 3, ..., and so on, and their use is mainly in manipulations and transformations of sets of equations within and between different coordinate systems. If you consider a force F with components fx, fy, fz, and you have an element of area whose NORMAL has components dSx, dSy, dSz, then fx itself has components acting on these three elements, and the PRESSURE of fx ALONE is denoted by its three components pxx, pxy, pxz Similarly fy will produce pressures pyx, pyy, pyz and fz will produce pressures pzx, pzy, pzz The product pxx.dSx gives the FORCE acting upon dSx by fx ALONE. It follows that: fx = pxx.dSx + pxy.dSy + pxz.dSz fy = pyz.dSx + pyy.dSy + pyz.dSz fz = pzx.dSx + pzy.dSy + pzz.dSz and the total STRESS F on the surface dS is F = fx + fy + fz which is given by the sum of the three equations (nine components) shown above. So we see that stress is not just a vector with three components (in three-dimensional space), but rather has NINE components in 3D space. Such a quantity is a TENSOR of rank 2. In general, if you are dealing with n-dimensional space, a tensor of rank 2 has n^2 components. Unlike a vector, whose components can be written in a single row or column, the components of a tensor of rank 2 will be written as a square array. In n-dimensional space, a tensor of rank 3 would have n^3 components. The need for a convenient notation allowing these arrays to be manipulated in an economical way is only too apparent. You must consult a textbook to see the notation, as it cannot be represented here in ASCII; but, for example, a capital letter with a couple of suffixes can be shorthand for a whole system of equations. -Doctor Anthony, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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