Curvilinear

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Unformatted text preview: ponents are with respect to the natural basis vectors or with respect to the physical basis vectors. To help maintain the distinction we use superscript numerals with the components in the natural basis and subscript letter (Latin and Greek) for components in the physical basis. Consider for example a vector v = vθ eθ = v 2 g 2 , then we have the relation vθ = rv 2 . (11) If for instance we have a vector v = vϕ eϕ + vθ eθ = v 2 g 2 + v 3 g 3 , then we have that vϕ = rv 2 vθ = r sin(ϕ)v 3 . (12) (13) Similar relations can be derived for tensor components. 3 Dual Basis Vectors When dealing with non-Cartesian coordinate systems one often introduces the so called dual (or contravariant) basis vectors; they are denoted by the symbol g k – note the raised index. The defining property of these basis vectors is that they are orthogonal to the first basis introduced; i.e. i g i · g j = δj , (14) i where δj is simply the Kronecker delta symbol. The the i index is raised so that it matches the other side of the equation. The meaning is still the same (1 if i = j and 0 otherwise). Another way of writing this is g k = (∂ z k /∂ xi )ei . 3 Note that for the usual Cartesian coordinates there is no difference between the dual basis and the regular basis. For the cylindrical system we have: cos(θ) − sin(θ)/r 0 1 2...
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This note was uploaded on 04/11/2013 for the course PHYS 105 taught by Professor Edgarknobloch during the Spring '10 term at University of California, Berkeley.

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