Coordinate Transformation jGeneral Tensors

9 Let us consider two bases bj(x\x3) and bj(xi, x2x3), Fig. 1.5. Each unit vector in one basis must be a linear combination of the vectors of the other basis bj- = apbp and bk = b\bg (1.20)

1.1 Vector

(summed on p and q respectively) where apP (subscript new, superscript old) and bk are the coefficients for the forward and backward changes respectively from b to b respectively. Explicitly

ei 1

r bi

e2

=

bi

e3

and and

Coordinate Transformation
Figure 1.5: Coordinate Transformation

10 The transformation must have the determinant of its Jacobian

dx1

dx1 dx1

dx1

dx2 dx3

j=

dx2

dx dx

= 0

dx1 dx3 dx1

dx2 dx3 dx3 dx3 dx2 dx3

different from zero (the superscript is a label and not an exponent).

11 It is important to note that so far, the coordinate systems are completely general and may be Cartesian, curvilinear, spherical or cylindrical.

0 0

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