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The Geometry of Physics by Theodore Frankel

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CHAPTER 2

Tensors and Exterior Forms

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IN Section 1.4b we considered the n-tuple of partial derivatives of a single function ∂F/∂xj and we noticed that this n-tuple does not transform in the same way as the n-tuple of components of a vector. These components ∂F/∂xj transform as a new type of “vector.” In this chapter we shall talk of the general notion of “tensor” that will include both notions of vector and a whole class of objects characterized by a transformation law generalizing 1.6. We shall, however, strive to define these objects and operations on them “intrinsically,” that is, in a basis-free fashion. We shall also be very careful in our ...

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