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Unformatted text preview: in other coordinates 1 Chapter 15 in other coordinates On a number of occasions we have noticed that del is geometrically determined it does not depend on a choice of coordinates for R n . This was shown to be true for f , the gradient of a function from R n to R (Section 2H). It was also verified for F , the divergence of a function from R n to R n (Section 14B). And most recently, we saw in Section 13G that it is true for F , the curl of a function from R 3 to R 3 . These three instances beg the question of how we might express in other coordinate systems for R n . A recent example of this is found in Section 13G, where a formula is given for F in terms of an arbitrary righthanded orthonormal frame for R 3 . We shall accomplish much more in this chapter. A very interesting book about , by Harry Moritz Schey, has the interesting title Div, Grad, Curl, And All That . A. Biorthogonal systems We begin with some elementary linear algebra. Consider an arbitrary frame { 1 , 2 ,..., n } for R n . We know of course that the Gram matrix is of great interest: G = ( i j ) . This is a symmetric positive definite matrix, and we shall denote its entries as g ij = i j . Of course, G is the identity matrix we have an orthonormal frame. We also form the matrix whose columns are the vectors i . Symbolically we write = ( 1 2 ... n ) . We know that is an orthogonal matrix we have an orthonormal frame (Problem 420). Now denote by the transpose of the inverse of , and express this new matrix in terms of its columns as = ( 1 2 ... n ) . We then have the matrix product t = t 1 . . . t n ( 1 ... n ) = ( i j ) . 2 Chapter 15 But since t equals the inverse of , we conclude that i j = ij . DEFINITION. Two frames { 1 ,..., n } and { 1 ,..., n } are called a biorthogonal system if i j = ij . Clearly, { 1 ,..., n } and { 1 ,..., n } form a biorthogonal system the frame { 1 ,..., n } is an orthonormal one. So the present definition should be viewed as a generalization of the concept of orthonormal basis. Also it is clear that the relation between the i s and the i s given here is completely symmetric. PROBLEM 151. Let a frame for R 2 be { ,a + b } , where of course b 6 = 0. Compute the corresponding { 1 , 2 } which produces a biorthogonal system. Sketch all four vectors on a copy of R 2 . PROBLEM 152. Let { 1 , 2 , 3 } be a frame for R 3 . Prove that the biorthogonal frame is given by 3 = 1 2 [ 1 , 2 , 3 ] etc ....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 Hatcher
 Calculus

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