tensor(deprecated)/Killing_eqns

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tensor

Killing_eqns

compute component expressions for Killings equations

	Calling Sequence
	Killing_eqns( T, coord, Cf2)

Parameters

T	-	symmetric covariant tensor
coord	-	list of coordinate names
Cf2	-	Christoffel symbols of the second kind

Description

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

•

The function Killing_eqns(T, coord, Cf2 ) computes the expressions for Killing's equations for each component of the totally symmetric covariant tensor T. Specifically, the symmetric part of the covariant derivative of T is computed and returned as a tensor_type. The components of T satisfy Killing's equations if all of the components of the result are zero. Note that the rank of the result is one more than that of T.

•	This routine is useful in two ways: first, as a means of verifying that a tensor satisfies Killing's equations, and second, as a way of generating the differential equations for any unknown components of a symmetric tensor which is to satisfy Killing's equations.

•	T must be of rank 1 or greater. If T is of second rank or more, the component array of T must use Maple's symmetric indexing function (since T must be symmetric).

•	Cf2 should be indexed using the cf2 indexing function provided by the tensor package. It can be computed using the Christoffel2 routine.

•

Simplification: This routine uses the `tensor/cov_diff/simp` and `tensor/lin_com/simp` routines for simplification purposes. The simplification routines are used indirectly by the symmetrize and cov_diff procedures as they are called by Killing_eqns. By default, `tensor/cov_diff/simp` and `tensor/lin_com/simp` are initialized to the `tensor/simp` routine. It is recommended that these routines be customized to suit the needs of the particular problem.

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

>	$with (tensor) &colon;$

Generate the Killing equation expressions for an arbitrary vector in the geometry of Euclidean 3-space using polar coordinates: First, compute the Christoffel symbols of the second kind:

>	$coord ≔ [r, θ, φ] &colon;$

>	$g_compts ≔ array (symmetric, sparse, 1 .. 3, 1 .. 3, [(1, 1) = 1, (2, 2) = r^{2}, (3, 3) = r^{2} {\sin (θ)}^{2}]) &colon;$

>	$g ≔ create ([- 1, - 1], eval (g_compts))$

$g ≔ table ([compts = [\begin{array}{c} 1 & 0 & 0 \\ 0 & r^{2} & 0 \\ 0 & 0 & r^{2} {\sin (θ)}^{2} \end{array}], index_char = [−1, −1]])$

(1)

>	$ginv ≔ invert (g,'detg') &colon;$

>	$d1g ≔ d1metric (g, coord) &colon;$ $d2g ≔ d2metric (d1g, coord) &colon;$

>	$Cf1 ≔ Christoffel1 (d1g) &colon;$

>	$Cf2 ≔ Christoffel2 (ginv, Cf1) &colon;$

Next, define the arbitrary vector field:

>	$V ≔ create ([- 1], array ([v1 (r, θ, φ), v2 (r, θ, φ), v3 (r, θ, φ)]))$

$V ≔ table ([compts = [\begin{array}{c} v1 (r, θ, φ) & v2 (r, θ, φ) & v3 (r, θ, φ) \end{array}], index_char = [−1]])$

(2)

Now compute the Killing equation expressions:

>	$KV ≔ Killing_eqns (V, coord, Cf2)$

$KV ≔ table ([compts = [\begin{array}{c} \frac{\partial}{\partial r} v1 (r, θ, φ) & \frac{(\frac{\partial}{\partial r} v2 (r, θ, φ)) r + (\frac{\partial}{\partial θ} v1 (r, θ, φ)) r - 2 v2 (r, θ, φ)}{2 r} & \frac{(\frac{\partial}{\partial r} v3 (r, θ, φ)) r + (\frac{\partial}{\partial φ} v1 (r, θ, φ)) r - 2 v3 (r, θ, φ)}{2 r} \\ \frac{(\frac{\partial}{\partial r} v2 (r, θ, φ)) r + (\frac{\partial}{\partial θ} v1 (r, θ, φ)) r - 2 v2 (r, θ, φ)}{2 r} & \frac{\partial}{\partial θ} v2 (r, θ, φ) + r v1 (r, θ, φ) & \frac{\frac{\partial}{\partial φ} v2 (r, θ, φ)}{2} - \cot (θ) v3 (r, θ, φ) + \frac{\frac{\partial}{\partial θ} v3 (r, θ, φ)}{2} \\ \frac{(\frac{\partial}{\partial r} v3 (r, θ, φ)) r + (\frac{\partial}{\partial φ} v1 (r, θ, φ)) r - 2 v3 (r, θ, φ)}{2 r} & \frac{\frac{\partial}{\partial φ} v2 (r, θ, φ)}{2} - \cot (θ) v3 (r, θ, φ) + \frac{\frac{\partial}{\partial θ} v3 (r, θ, φ)}{2} & \frac{\partial}{\partial φ} v3 (r, θ, φ) + r {\sin (θ)}^{2} v1 (r, θ, φ) + \frac{\sin (2 θ) v2 (r, θ, φ)}{2} \end{array}], index_char = [−1, −1]])$

(3)

Now try it for an arbitrary symmetric 0, 2-tensor:

>	$t ≔ array (symmetric, 1 .. 3, 1 .. 3) &colon;$

>	$for i to 3 do for j from i to 3 do t [i, j] ≔ cat ('t', i, j) (r, θ, φ) end do end do &semi;$ $T ≔ create ([- 1, - 1], eval (t))$

$t33 (r, θ, φ)$

$T ≔ table ([compts = [\begin{array}{c} t11 (r, θ, φ) & t12 (r, θ, φ) & t13 (r, θ, φ) \\ t12 (r, θ, φ) & t22 (r, θ, φ) & t23 (r, θ, φ) \\ t13 (r, θ, φ) & t23 (r, θ, φ) & t33 (r, θ, φ) \end{array}], index_char = [−1, −1]])$

(4)

>	$KT ≔ Killing_eqns (T, coord, Cf2)$

$KT ≔ table ([compts = array (symmetric, 1 .. 3, 1 .. 3, 1 .. 3, [(1, 1, 1) = \frac{\partial}{\partial r} t11 (r, θ, φ), (1, 1, 2) = \frac{2 (\frac{\partial}{\partial r} t12 (r, θ, φ)) r + (\frac{\partial}{\partial θ} t11 (r, θ, φ)) r - 4 t12 (r, θ, φ)}{3 r}, (1, 1, 3) = \frac{2 (\frac{\partial}{\partial r} t13 (r, θ, φ)) r + (\frac{\partial}{\partial φ} t11 (r, θ, φ)) r - 4 t13 (r, θ, φ)}{3 r}, (1, 2, 1) = \frac{2 (\frac{\partial}{\partial r} t12 (r, θ, φ)) r + (\frac{\partial}{\partial θ} t11 (r, θ, φ)) r - 4 t12 (r, θ, φ)}{3 r}, (1, 2, 2) = \frac{2 r^{2} t11 (r, θ, φ) + (\frac{\partial}{\partial r} t22 (r, θ, φ)) r + 2 (\frac{\partial}{\partial θ} t12 (r, θ, φ)) r - 4 t22 (r, θ, φ)}{3 r}, (1, 2, 3) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (1, 3, 1) = \frac{2 (\frac{\partial}{\partial r} t13 (r, θ, φ)) r + (\frac{\partial}{\partial φ} t11 (r, θ, φ)) r - 4 t13 (r, θ, φ)}{3 r}, (1, 3, 2) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (1, 3, 3) = \frac{2 r^{2} {\sin (θ)}^{2} t11 (r, θ, φ) + \sin (2 θ) t12 (r, θ, φ) r + 2 (\frac{\partial}{\partial φ} t13 (r, θ, φ)) r - 4 t33 (r, θ, φ) + (\frac{\partial}{\partial r} t33 (r, θ, φ)) r}{3 r}, (2, 1, 1) = \frac{2 (\frac{\partial}{\partial r} t12 (r, θ, φ)) r + (\frac{\partial}{\partial θ} t11 (r, θ, φ)) r - 4 t12 (r, θ, φ)}{3 r}, (2, 1, 2) = \frac{2 r^{2} t11 (r, θ, φ) + (\frac{\partial}{\partial r} t22 (r, θ, φ)) r + 2 (\frac{\partial}{\partial θ} t12 (r, θ, φ)) r - 4 t22 (r, θ, φ)}{3 r}, (2, 1, 3) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (2, 2, 1) = \frac{2 r^{2} t11 (r, θ, φ) + (\frac{\partial}{\partial r} t22 (r, θ, φ)) r + 2 (\frac{\partial}{\partial θ} t12 (r, θ, φ)) r - 4 t22 (r, θ, φ)}{3 r}, (2, 2, 2) = \frac{\partial}{\partial θ} t22 (r, θ, φ) + 2 r t12 (r, θ, φ), (2, 2, 3) = \frac{\frac{\partial}{\partial φ} t22 (r, θ, φ)}{3} - \frac{4 \cot (θ) t23 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial θ} t23 (r, θ, φ)}{3} + \frac{2 t13 (r, θ, φ) r}{3}, (2, 3, 1) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (2, 3, 2) = \frac{\frac{\partial}{\partial φ} t22 (r, θ, φ)}{3} - \frac{4 \cot (θ) t23 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial θ} t23 (r, θ, φ)}{3} + \frac{2 t13 (r, θ, φ) r}{3}, (2, 3, 3) = \frac{2 r {\sin (θ)}^{2} t12 (r, θ, φ)}{3} + \frac{2 \sin (θ) \cos (θ) t22 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial φ} t23 (r, θ, φ)}{3} - \frac{4 \cot (θ) t33 (r, θ, φ)}{3} + \frac{\frac{\partial}{\partial θ} t33 (r, θ, φ)}{3}, (3, 1, 1) = \frac{2 (\frac{\partial}{\partial r} t13 (r, θ, φ)) r + (\frac{\partial}{\partial φ} t11 (r, θ, φ)) r - 4 t13 (r, θ, φ)}{3 r}, (3, 1, 2) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (3, 1, 3) = \frac{2 r^{2} {\sin (θ)}^{2} t11 (r, θ, φ) + \sin (2 θ) t12 (r, θ, φ) r + 2 (\frac{\partial}{\partial φ} t13 (r, θ, φ)) r - 4 t33 (r, θ, φ) + (\frac{\partial}{\partial r} t33 (r, θ, φ)) r}{3 r}, (3, 2, 1) = \frac{- 2 \cot (θ) t13 (r, θ, φ) r + (\frac{\partial}{\partial φ} t12 (r, θ, φ)) r - 4 t23 (r, θ, φ) + (\frac{\partial}{\partial θ} t13 (r, θ, φ)) r + (\frac{\partial}{\partial r} t23 (r, θ, φ)) r}{3 r}, (3, 2, 2) = \frac{\frac{\partial}{\partial φ} t22 (r, θ, φ)}{3} - \frac{4 \cot (θ) t23 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial θ} t23 (r, θ, φ)}{3} + \frac{2 t13 (r, θ, φ) r}{3}, (3, 2, 3) = \frac{2 r {\sin (θ)}^{2} t12 (r, θ, φ)}{3} + \frac{2 \sin (θ) \cos (θ) t22 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial φ} t23 (r, θ, φ)}{3} - \frac{4 \cot (θ) t33 (r, θ, φ)}{3} + \frac{\frac{\partial}{\partial θ} t33 (r, θ, φ)}{3}, (3, 3, 1) = \frac{2 r^{2} {\sin (θ)}^{2} t11 (r, θ, φ) + \sin (2 θ) t12 (r, θ, φ) r + 2 (\frac{\partial}{\partial φ} t13 (r, θ, φ)) r - 4 t33 (r, θ, φ) + (\frac{\partial}{\partial r} t33 (r, θ, φ)) r}{3 r}, (3, 3, 2) = \frac{2 r {\sin (θ)}^{2} t12 (r, θ, φ)}{3} + \frac{2 \sin (θ) \cos (θ) t22 (r, θ, φ)}{3} + \frac{2 \frac{\partial}{\partial φ} t23 (r, θ, φ)}{3} - \frac{4 \cot (θ) t33 (r, θ, φ)}{3} + \frac{\frac{\partial}{\partial θ} t33 (r, θ, φ)}{3}, (3, 3, 3) = \frac{\partial}{\partial φ} t33 (r, θ, φ) + 2 r {\sin (θ)}^{2} t13 (r, θ, φ) + \sin (2 θ) t23 (r, θ, φ)]), index_char = [−1, −1, −1]])$

(5)

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