TensorDate: 10/01/97 at 14:13:17 From: MIKE VARELA Subject: TENSORS/DEFINITION Dear Dr. Math, Is there a layman's definition for a tensor? And is there a simple workable example to illustrate this definition? Thank you! Mikey V. Date: 10/01/97 at 17:24:06 From: Doctor Anthony Subject: Re: TENSORS/DEFINITION 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 = 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 which 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 Check out our web site! http://mathforum.org/dr.math/ Date: 10/02/97 at 09:51:39 From: Mike Varela Subject: Re: TENSORS/DEFINITION Dr. Anthony, That was the clearest definition I ever heard for tensors. Thanks! However, I don't understand why 3D surfaces need to be defined by their normals dSx, dSy, dSz. Does it mean that stress F, with its fx, fy and fz components, acting on the xyz components of the surface, is being looked at in terms of its perpendicular influence, or pressure? Which would mean in that case that F is ALIGNED (parallel) to the dSx, dSy aND dSz NORMALS to produce the pressure on the xyz-components of the surface in question? If so, seeing that is how a real force influences the surface of an object, then I can see why it's important to align such a matrix in terms of the surface's normals. Mikey V. Date: 10/02/97 at 14:21:21 From: Doctor Anthony Subject: Re: TENSORS/DEFINITION Hello, The items pxx, pxy, pxz are forces per unit area (hence the term pressure), and these are the pressures arising from fx alone on the components of area, dSx, dSy and dSz. In all this work it is much more convenient to have forces, velocities, momentums, areas, torsions and almost any physical property you can think of represented in component form, and because of this the number of possible components and possible combinations of components expands so rapidly. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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