Use of TensorsDate: 7/23/96 at 0:11:37 From: STU R U? Subject: Use of Tensors Hi! Could you please explain the uses and properties of tensors? Date: 7/23/96 at 17:52:29 From: Doctor Anthony Subject: Re: Use of Tensors We could start by saying that a SCALAR is a TENSOR whose RANK is zero, and a VECTOR is a TENSOR whose RANK is one. What we shall need to see is what is meant by a tensor of rank 2, or 3 etc. As you can see, TENSOR is an inclusive term, a generalization of the concept of vector. To illustrate a tensor of rank 2, imagine a plane surface area with a force acting on it. The total effect depends on two things, the magnitude and direction of the force, and the size of the area and its orientation. In fact this latter property can be represented uniquely by a vector of magnitude proportional to the size of the area, and in a direction NORMAL to the area. So the effect of the force upon the surface depends on two vectors and is called a TENSOR of RANK TWO. In fact, if you consider components of force and each of these components acting on each component of the area vector, there are nine terms in all, which can be displayed as an array, representing the total stress, and this quantity is the tensor of rank two. Tensors can thus be represented by arrays, and manipulated in a manner reminiscent of matrix manipulation. They have particular importance in problems involving invariance in changes of a coordinate system, and, famously, were used by Einstein in deriving his law of gravitation in General Relativity. The notation which uses double suffixes can not be recommended in ASCII, and I think you will need to dig further into this topic with the aid of a textbook on advanced topics in vectors. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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