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# lec10-1 - Vector operators in curvilinear coordinate...

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Unformatted text preview: Vector operators in curvilinear coordinate systems • In a Cartesian system, take x 1 = x , x 2 = y , and x 3 = z , then an element of arc length ds 2 is, ds 2 = dx 2 1 + dx 2 2 + dx 2 3 • In a general system of coordinates, we still have x 1 , x 2 , and x 3 • For example, in cylindrical coordinates, we have x 1 = r , x 2 = θ , and x 3 = z • We have already shown how we can write ds 2 in cylindrical coordinates, ds 2 = dr 2 + r 2 d θ + dz 2 = dx 2 1 + x 2 1 dx 2 2 + dx 2 3 • We write this in a general form, with h i being the scale factors ds 2 = h 2 1 dx 2 1 + h 2 2 dx 2 2 + h 2 3 dx 2 3 • We see then for cylindrical coordinates, h 1 = 1, h 2 = r , and h 3 = 1 Curvilinear coordinates • For an vector displacement ~ ds ~ ds = ˆ e 1 h 1 dx 1 + ˆ e 2 h 2 dx 2 + ˆ e 3 h 3 dx 3 • Back to our example of cylindrical coordiantes, ˆ e 1 = ˆ e r , ˆ e 2 = ˆ e θ , and ˆ e 3 = ˆ e z , and ~ ds = ˆ e r dr + ˆ e θ rd θ + ˆ e z dz • These are orthogonal systems, but it would not have to be! ds 2 = 3 X i =1 3 X j =1 g ij dx i dx i • The g ij is the metric tensor, and for an orthogonal system it is diagonal with g i = h 2 i Vector operators in general curvilinear coordinates • Recall the directional derivative d φ ds along ~ u , where ~ u was a unit vector d φ ds = ∇ φ · ~ u • Now the ~ u becomes the unit vectors in an orthogonal system, for example in cylindrical coordinates • Now we recall that ds 2 = ds 2 = h 2 1 dx 2 1 + h 2 2 dx 2 2 + h 2 3 dx 2 3 • Let’s take a cylindrical system, first consider ~ u = ˆ e r , then ds = dr ~ ∇ φ ( r ,θ, z ) · ˆ e r = ∂φ ∂ r Vector operators in general curvilinear coordinates • Next ~ u = ˆ e r θ , then ds = rd θ ( h 2 = r ) ~ ∇ φ ( r ,θ, z ) · ˆ e θ = 1 r ∂φ ∂θ • It is also easy to show, ~ ∇ φ ( r ,θ, z ) · ˆ e z = ∂φ ∂ z • Now that we have the projections, we can find ~ ∇ φ in cylindrical coordinates, ~ ∇ φ = ∂φ ∂ r ˆ e r + 1 r ∂φ ∂θ ˆ e θ + ∂φ ∂ z ˆ e z Gradient in curvilinear (orthogonal) coordinate system...
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lec10-1 - Vector operators in curvilinear coordinate...

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