By Werner Hildbert Greub, Stephen Halperin, James Van Stone

ISBN-10: 0123027039

ISBN-13: 9780123027030

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**Example text**

Let y D y i Ag i be an arbitrary vector in En . 130). Then, Ax D y i Ag i D y which implies that the tensor A is inverse to A 1 . 131) implies the uniqueness of the inverse. Indeed, if A 1 and A 1 are two distinct tensors both inverse to A then there exists at least one vector y 2 En such that A 1 y ¤ A 1 y. 131) into account we immediately come to the contradiction. 130) x D B 1 A 1 y; 8x 2 En : On the basis of transposition and inversion one defines the so-called orthogonal tensors. 73) is orthogonal.

Its elements are second-order tensors that can be treated as vectors in En with all the operations specific for vectors such as summation, multiplication by a scalar or a scalar product (the latter one will be defined for second-order tensors in Sect. 10). 9 Special Operations with Second-Order Tensors 21 for second-order tensors one can additionally define some special operations as for example composition, transposition or inversion. Composition (simple contraction). Let A; B 2 Linn be two second-order tensors.

T/ : dt dt dt 2. t/ : dt dt dt M. 5) 35 36 2 Vector and Tensor Analysis in Euclidean Space 3. t/ W : dt dt dt 4. 8) 5. 9) 6. t/ D : dt du dt dt du dt 7. 12) dt @u dt @v dt where @=@u denotes the partial derivative. 0 @u s The above differentiation rules can be verified with the aid of elementary differential calculus. 9) we proceed as follows. 1. x 1 ; x 2 ; : : : ; x n /. These numbers are called coordinates of the corresponding vectors. i D 1; 2; : : : ; n/. r/ and r D r x 1 ; x 2 ; : : : ; x n are sufficiently differentiable.

### Connections, curvature, and cohomology by Werner Hildbert Greub, Stephen Halperin, James Van Stone

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