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Unformatted text preview: Tensors You can’t walk across a room without using a tensor (the pressure tensor). You can’t align the wheels on your car without using a tensor (the inertia tensor). You definitely can’t understand Einstein’s theory of gravity without using tensors (many of them). This subject is often presented in the same language in which it was invented in the 1890’s, expressing it in terms of transformations of coordinates and saturating it with such formidable-looking combinations as ∂x i /∂ ¯ x j . This is really a sideshow to the subject, one that I will steer around, though a connection to this aspect appears in section 12.8 . Some of this material overlaps that of chapter 7, but I will extend it in a different direction. The first examples will then be familiar. 12.1 Examples A tensor is a particular type of function. Before presenting the definition, some examples will clarify what I mean. Start with a rotating rigid body, and compute its angular momentum. Pick an origin and assume that the body is made up of N point masses m i at positions described by the vectors ~ r i ( i = 1 , 2 ,...,N ). The angular velocity vector is ~ω . For each mass the angular momentum is ~ r i × ~ p i = ~ r i × ( m i ~v i ) . The velocity ~v i is given by ~ω × ~ r i and so the angular momentum of the i th particle is m i ~ r i × ( ~ω × ~ r i ) . The total angular momentum is therefore m 1 m 2 m 3 ~ω ~ L = N X i =1 m i ~ r i × ( ~ω × ~ r i ) . (12 . 1) The angular momentum, ~ L , will depend on the distribution of mass within the body and upon the angular velocity. Write this as ~ L = I ( ~ω ) , where the function I is called the tensor of inertia. For a second example, take a system consisting of a mass suspended by six springs. At equilibrium the springs are perpendicular to each other. If now a (small) force ~ F is applied to the mass it will undergo a displacement ~ d . Clearly, if ~ F is along the direction of any of the springs then the displacement ~ d will be in the same direction as ~ F . Suppose however that ~ F is halfway between the k 1 and k 2 springs, and further that the spring k 2 was taken from a railroad locomotive while k 1 is a watch spring. Obviously in this case ~ d will be mostly in the x direction ( k 1 ) and is not aligned with ~ F . In any case there is a relation between ~ d and ~ F , ~ d = f ( ~ F ) . (12 . 2) The function f is a tensor. In both of these examples, the functions involved were vector valued functions of vector variables . They have the further property that they are linear functions, i.e. if α and β are real numbers, I ( α~ω 1 + β~ω 2 ) = αI ( ~ω 1 ) + βI ( ~ω 2 ) , f ( α ~ F 1 + β ~ F 2 ) = αf ( ~ F 1 ) + βf ( ~ F 2 ) These two properties are the first definition of a tensor. (A generalization will come later.) There’s a point here that will probably cause some confusion. Notice that in the equation ~ L = I ( ~ω ) , James Nearing, University of Miami 1 12—Tensors 2 the tensor is the function...
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