Prelim 1 oct. 2007 - A&EP 321 Mathematical Physics...

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A&EP 321 Mathematical Physics Prelim #1 October 2, 2007 7:30-9:30 p.m. CLOSED BOOK NO CALCULATORS
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1. Let A and B be eigenvectors of the tensor T with eigenvalues α and β , 20% respectively T · A = α A T · B = β B where all tensosr elements, all eigenvector components and the eigenvalues are pure real. a. What are the conditions on the elements of T so that A · T · B = B · T · A ? b. If the elements of T satisfy the condition above show that the eigenvectors are orthogonal. c. Find the eigenvalues and eigenvectors associated with T = T ij ˆe i j with [ T ij ]= ± 14 32 ² . Are the eigenvectors orthogonal? Comment. 2. Develop a vector-tensor identity for the following expression 15% ( A × B ) × C where A and C are second rank tensors and B is a vector. 1
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3. A two dimensional curvilinear coordinate system has coordinates u and v 25% that are related to the Cartesian coordinates as follows: u = x 2 + y 2 2 x v = x 2 + y 2 2 y . a. Graph the lines of constant u and constant v in the xy -plane.
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This note was uploaded on 11/18/2008 for the course A&EP 321 taught by Professor Kusse during the Fall '07 term at Cornell University (Engineering School).

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Prelim 1 oct. 2007 - A&EP 321 Mathematical Physics...

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