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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
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.
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)
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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.
- Spring '10