Definition of a TensorDate: 05/20/2001 at 10:00:41 From: M Innes Subject: Clear concise definition of a tensor If a vector can be described as "any quantity having magnitude and direction" why can't a tensor be defined as "any..." in just a couple of lines? Many thanks. Date: 05/20/2001 at 17:32:42 From: Doctor Anthony Subject: Re: Clear concise definition of a tensor 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 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 = pyx.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 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 that allows these arrays to be manipulated in an economical way is only too apparent. You must consult a textbook to see the notation, it cannot be represented here in ASCII, but a capital letter with a couple of suffixes can be shorthand for a whole system of equations. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ Date: 05/21/2001 at 20:59:18 From: Doctor Schwa Subject: Re: Clear concise definition of a tensor Hi Mark, My approach would go like this: You know what a vector is. Let's think of that as being a column matrix: an n by 1 matrix. Then a covector will be a row matrix: a 1 by n matrix. Another way to define a covector is that it's a linear function that takes each vector to a number: in other words, you can multiply a covector by any vector and get a number. You could write that as c(v) = a number. A matrix takes both a covector and a vector to produce a number: cMv (row times matrix times column) will produce a number. Matrix multiplication also has certain associative/distributive laws, and some properties of how it acts when one of those vectors (c or v) are multiplied by a scalar. Those are called "linearity rules" or "linear properties" or things like that. Furthermore, if you "feed" the matrix only a vector, multiplying Mv, you'd be left with a vector. If you "feed" it only a covector, multiplying cM, you're left with a covector, still "hungry" for a vector to produce a final numerical answer. A tensor is like a matrix, but it can have more than two dimensions: it takes some collection of covectors and/or vectors, and produces a number, following the same kind of linearity rules as you'd expect from matrix multiplication. And if you "feed" it less than its full complement of desired vectors and covectors, you'll be left with a tensor that's still "hungry" for a few things. It only makes sense to equate tensors that have "hunger" for the same things, just like you can only equate vectors with other vectors and matrices with other matrices. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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