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Bending of curved tubes by Hovgaard W.

By Hovgaard W.

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We define ´¾µ , Ù´¾µ and ´¾µ as ¾ ¼ ¿ ´½µ ¾¾ ´½µ ¿¾ ´¾µ .. ¾ Ù´¾µ ¼ ´½µ ¾¾ ´½µ Ѿ ´¾µ where ¾ ¦ È ´¾µ ´½µ ¿¾ ¿ ¾ ´¾µ ¾ ´¾µ ¾ ¼ ´¾µ .. .. ¼ ´½µ Ѿ ´¾µ . Let È ´¾µ be defined by Á · Ù´¾µ ´Ù´¾µ µÌ ´ ´¾µµÌ Ù´¾µ ¼ ¿ ¾ ½ ¼ ¡¡¡ ¼ ¿ ¼ Ⱦ¾´¾µ ¡ ¡ ¡ Ⱦ´¾µÑ .. .. .. .. ´¾µ ¼ ÈÑ´¾µ¾ ¡ ¡ ¡ ÈÑÑ 26 2 Linear Algebra and Preliminaries We see that È ´¾µ is an orthogonal matrix, for which all the elements of the first row and column are zero except for ´½ ½µ-element. Thus pre-multiplying ´½µ by È ´¾µ yields ¿ ¾ ´½µ ½ È ´¾µ ´½µ È ´¾µ È ´½µ ´½µ ½¾ ´¾µ ¾ ´¾µ ´¾µ ½¿ ¡ ¡ ¡ ´¾µ ¾¿ ¡ ¡ ¡ ´¾µ ¿¿ ¡ ¡ ¡ ´¾µ ½Ò ´¾µ ¾Ò ´¾µ ¿Ò ´¾µ ¿ ¡¡¡ ´¾µ ..

If there is no confusion, we simply write Á , deleting the subscript denoting the dimension. The inverse of a square matrix Ì µ ½ . A matrix is denoted by ½ . We also use Ì to denote ´ ½ µÌ ´ Ì Ñ is called a symmetric matrix. If a matrix ¾ Ê ¢Ò with Ñ Ò satisfying Ì satisfies ÁÒ , it is called an orthogonal matrix. 1) Define ´ · Ì µ ¾. Then, we have ÜÌ Ü ÜÌ Ü. Thus it is assumed without loss of generality that is symmetric in defining a quadratic form. If ÜÌ Ü ¼ Ü ¼, then is positive definite, and is written as ¼.

41) It therefore follows that any row vector of À below the ´Ð ·½µth row can be expressed in terms of a linear combination of the Ð preceding row vectors, and hence in terms of linearly independent first Ð row vectors. 40). 40) holds. Then, all the rows (columns) of À are expressed in terms of linear combinations of the first Ö rows (columns). Thus all the minors of À whose orders are greater than Ö are zero, and À has rank Ö at most. 40) is satisfied with a smaller value of Ö. This contradicts the second condition of the lemma.

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