For the basis vectors we will introduce for two types

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Unformatted text preview: 3 ) = (r, ϕ, θ) in the spherical case. For the basis vectors we will introduce for two types of basis vectors. The natural basis vectors and 1 x3 x3 ϕ r x2 x2 z r θ θ x1 x1 Figure 1: Definition of the cylindrical and spherical coordinate systems. the physical basis vectors. Both bases are orthogonal but the physical basis has the additional property of orthonormality. The basic definitions are ∂ xi ei ∂zk gk . = ￿g k ￿ gk = (5) ek (6) To differentiate between the physical basis vectors and the usual Cartesian ones we typically write er , eθ , · · · etc. For the cylindrical coordinate system one has: cos(θ) −r sin(θ) 0 sin(θ) r cos(θ) 0 g1 → g2 → g3 → (7) 0 0 1 and 1 eθ = g 2 ez = g 3 . (8) r Note that the components have been expressed in the standard orthonormal Cartesian basis. For the spherical system one has that sin(ϕ) cos(θ) r cos(ϕ) cos(θ) g 1 → sin(ϕ) sin(θ) g 2 → r cos(ϕ) sin(θ) cos(ϕ) −r sin(ϕ) (9) −r sin(ϕ) sin(θ) g 3 → r sin(ϕ) cos(θ) 0 er = g 1 2 and er = g 1 2.1 1 eϕ = g 2 r eθ = 1 g. r sin(ϕ) 3 (10) Physical and Natural Components As with all bases we can express the components of vectors and tensors with respect to our new curvilinear bases. In this regard, it is very important with the curvilinear coordinates to know whether or not the com...
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