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5.3 Tensor calculus in polar coordinates |
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Divergence and Laplacian |
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Before doing any more theory, let us link this up with things we have seen before. In |
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Cartesian coordinates the divergence of a vector Vα is Vα ,α . This is the scalar obtained |
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by contracting Vα ,β on its two indices. Since contraction is a frame-invariant operation, |
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the divergence of V can be calculated in other coordinates {xμ } also by contracting the |
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components of V on their two indices. This results in a scalar with the value Vα ;α . It is |
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important to realize that this is the same number as Vα ,α in Cartesian coordinates: |
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Vα ,α ≡ Vβ ;β , |
(5.54) |
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where unprimed indices refer to Cartesian coordinates and primed refer to the arbitrary |
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system. |
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For polar coordinates (dropping primes for convenience here)
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∂Vα |
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Vα ;α = |
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+ α μα Vμ. |
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∂xα |
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Now, from Eq. (5.45) we can calculate |
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θ r+ |
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θ θ θ= |
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α θ α |
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r |
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0. |
(5.55) |
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α rα |
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rrr |
θ rθ |
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1/r, |
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= |
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+ |
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= |
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Therefore we have |
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∂Vr |
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∂Vθ |
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1 |
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Vα ;α = |
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+ |
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+ |
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Vr, |
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∂r |
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∂θ |
r |
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= |
1 ∂ |
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(rVr) + |
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∂ |
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Vθ . |
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(5.56) |
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r |
∂r |
∂θ |
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This may be a familiar formula to the student. What is probably more familiar is the Laplacian, which is the divergence of the gradient. But we only have the divergence of vectors, and the gradient is a one-form. Therefore we must first convert the one-form to a vector. Thus, given a scalar φ, we have the vector gradient (see Eq. (5.53) and the last part of § 5.2 above) with components (φ,r, φ,θ /r2). Using these as the components of the vector in the
divergence formula, Eq. (5.56) gives |
φ = r ∂r |
r ∂r |
+ r2 |
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∂θ 2 . |
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· φ := 2 |
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(5.57) |
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1 ∂ |
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∂φ |
1 |
∂2φ |
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This is the Laplacian in plane polar coordinates. It is, of course, identically equal to |
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2 |
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∂2φ |
∂2φ |
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φ = |
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(5.58) |
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∂x2 |
∂y2 |
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Derivatives of one -forms and tensors of higher t ypes
˜
Since a scalar φ depends on no basis vectors, its derivative dφ is the same as its covariant derivative φ. We shall almost always use the symbol φ. To compute the derivative of a one-form (which as for a vector won’t be simply the derivatives of its components), we use
130 |
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Preface to curvature |
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˜ |
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the property that a one-form and a vector give a scalar. Thus, if p is a one-form and V is an |
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arbitrary vector, then for fixed β, |
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˜ |
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β V is a vector, and |
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≡ |
φ |
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β p is also a one-form, |
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p, V |
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is a scalar. In any (arbitrary) coordinate system this scalar is just |
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φ = pα Vα . |
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(5.59) |
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Therefore β φ is, by the product rule for derivatives, |
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∂pα |
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∂Vα |
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β φ = φ,β = |
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Vα |
+ pα |
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(5.60) |
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∂xβ |
∂xβ |
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But we can use Eq. (5.50) to replace ∂Vα /∂xβ in favor of Vα ;β , which are the components |
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of β V: |
∂pα |
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β φ = |
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Vα + pα Vα ;β − pα Vμ α μβ . |
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(5.61) |
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∂xβ |
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Rearranging terms, and relabeling dummy indices in the term that contains the Christoffel symbol, gives
β φ = |
∂pα |
− pμ μαβ |
Vα + pα Vα ;β . |
(5.62) |
∂xβ |
Now, every term in this equation except the one in parentheses is known to be the compo-
nent of a tensor, for an arbitrary vector field V. Therefore, since multiplication and addition of components always gives new tensors, it must be true that the term in parentheses is also the component of a tensor. This is, of course, the covariant derivative of p˜:
( β p˜)α := ( p˜)αβ := pα;β = pα,β − pμ μαβ . |
(5.63) |
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Then Eq. (5.62) reads |
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β (pα Vα ) = pα;β Vα + pα Vα ;β . |
(5.64) |
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Thus covariant differentiation obeys the same sort of product rule as Eq. (5.60). It must do this, since in Cartesian coordinates is just partial differentiation of components, so Eq. (5.64) reduces to Eq. (5.60).
Let us compare the two formulae we have, Eq. (5.50) and Eq. (5.63):
Vα ;β = Vα ,β + Vμ α μβ , pα;β = pα,β − pμ μαβ .
There are certain similarities and certain differences. If we remember that the derivative index β is the last one on , then the other indices are the only ones they can be without raising and lowering with the metric. The only thing to watch is the sign difference. It may help to remember that α μβ was related to derivatives of the basis vectors, for then it is reasonable that − μαβ be related to derivatives of the basis one-forms. The change in sign means that the basis one-forms change ‘oppositely’ to basis vectors, which makes sense