tensor(deprecated)/d1metric - Maple Help

Home : Support : Online Help : tensor(deprecated)/d1metric

tensor

 d1metric
 compute the first partial derivatives of the covariant metric tensor components

 Calling Sequence d1metric( g, coord)

Parameters

 g - covariant metric tensor coord - list of names representing the coordinate variables

Description

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

 • Given the coordinate variables, coord, and the covariant metric tensor, g, d1metric( g, coord ) computes the partial derivatives of g, which will be a new tensor_type of rank three and which uses the cf1 indexing function provided by the tensor package to implement the symmetry in the first two indices of the partials.  Thus, the symmetry of the metric components is preserved by the d1metric function.
 • g must be a rank two tensor_type with character [-1,-1] and must use Maple's symmetric indexing function for its component array.
 • The extra index due to differentiation is of covariant character, by convention. Thus, the index_char field of the result is [-1, -1, -1].
 • Indexing Function: The cf1 indexing function is used to implement the symmetry properties of the first partials of the covariant metric tensor components (it is also used for the Christoffel symbols of the first kind, hence the name cf1). Specifically, it implements the symmetry in the first and second indices of a three index quantity.
 • Simplification:  This routine uses the tensor/d1metric/simp routine for simplification purposes.  The simplification routine is applied to each component of result after it is computed.  By default, tensor/d1metric/simp is initialized to the tensor/simp routine.  It is recommended that the tensor/d1metric/simp routine be customized to suit the needs of the particular problem. For example, if the metric component functions are not fully known but are known to satisfy certain differential equations, the tensor/d2metric/simp routine could be used to make substitutions for the partial derivatives of these functions.
 • To compute the second partials of the metric components while preserving the symmetry in the metric components and in the second partials, use tensor[d2metric]. For computing the partial derivatives of the components of an arbitrary tensor, use tensor[partial_diff].
 • This function is part of the tensor package, and so can be used in the form d1metric(..) only after performing the command with(tensor) or with(tensor, d1metric).  The function can always be accessed in the long form tensor[d1metric](..).

Examples

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

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

Define the coordinate variables and the covariant components of the Schwarzchild metric.

 > $\mathrm{coord}≔\left[t,r,\mathrm{\theta },\mathrm{\phi }\right]:$
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..4,1..4\right):$
 > $\mathrm{g_compts}\left[1,1\right]≔1-\frac{2m}{r}:$$\mathrm{g_compts}\left[2,2\right]≔-\frac{1}{\mathrm{g_compts}\left[1,1\right]}:$
 > $\mathrm{g_compts}\left[3,3\right]≔-{r}^{2}:$$\mathrm{g_compts}\left[4,4\right]≔-{r}^{2}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}:$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{1}{-}\frac{{2}{}{m}}{{r}}& {0}& {0}& {0}\\ {0}& {-}\frac{{1}}{{1}{-}\frac{{2}{}{m}}{{r}}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}& {0}\\ {0}& {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (1)

Compute the first partials of g:

 > $\mathrm{d1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right)$
 ${\mathrm{d1g}}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}{array}{}\left({\mathrm{cf1}}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}\left[\left({1}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{2}\right){=}\frac{{2}{}{m}}{{{r}}^{{2}}}{,}\left({1}{,}{1}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{4}\right){=}{0}{,}\left({1}{,}{2}{,}{1}\right){=}{0}{,}\left({1}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{2}{,}{3}\right){=}{0}{,}\left({1}{,}{2}{,}{4}\right){=}{0}{,}\left({1}{,}{3}{,}{1}\right){=}{0}{,}\left({1}{,}{3}{,}{2}\right){=}{0}{,}\left({1}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{3}{,}{4}\right){=}{0}{,}\left({1}{,}{4}{,}{1}\right){=}{0}{,}\left({1}{,}{4}{,}{2}\right){=}{0}{,}\left({1}{,}{4}{,}{3}\right){=}{0}{,}\left({1}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{1}{,}{2}\right){=}{0}{,}\left({2}{,}{1}{,}{3}\right){=}{0}{,}\left({2}{,}{1}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{2}\right){=}\frac{{2}{}{m}}{{\left({-}{r}{+}{2}{}{m}\right)}^{{2}}}{,}\left({2}{,}{2}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{4}\right){=}{0}{,}\left({2}{,}{3}{,}{1}\right){=}{0}{,}\left({2}{,}{3}{,}{2}\right){=}{0}{,}\left({2}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{3}{,}{4}\right){=}{0}{,}\left({2}{,}{4}{,}{1}\right){=}{0}{,}\left({2}{,}{4}{,}{2}\right){=}{0}{,}\left({2}{,}{4}{,}{3}\right){=}{0}{,}\left({2}{,}{4}{,}{4}\right){=}{0}{,}\left({3}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{1}{,}{2}\right){=}{0}{,}\left({3}{,}{1}{,}{3}\right){=}{0}{,}\left({3}{,}{1}{,}{4}\right){=}{0}{,}\left({3}{,}{2}{,}{1}\right){=}{0}{,}\left({3}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{2}{,}{3}\right){=}{0}{,}\left({3}{,}{2}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{2}\right){=}{-}{2}{}{r}{,}\left({3}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{4}\right){=}{0}{,}\left({3}{,}{4}{,}{1}\right){=}{0}{,}\left({3}{,}{4}{,}{2}\right){=}{0}{,}\left({3}{,}{4}{,}{3}\right){=}{0}{,}\left({3}{,}{4}{,}{4}\right){=}{0}{,}\left({4}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{1}{,}{2}\right){=}{0}{,}\left({4}{,}{1}{,}{3}\right){=}{0}{,}\left({4}{,}{1}{,}{4}\right){=}{0}{,}\left({4}{,}{2}{,}{1}\right){=}{0}{,}\left({4}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{2}{,}{3}\right){=}{0}{,}\left({4}{,}{2}{,}{4}\right){=}{0}{,}\left({4}{,}{3}{,}{1}\right){=}{0}{,}\left({4}{,}{3}{,}{2}\right){=}{0}{,}\left({4}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{3}{,}{4}\right){=}{0}{,}\left({4}{,}{4}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{2}\right){=}{-}{2}{}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{,}\left({4}{,}{4}{,}{3}\right){=}{-}{2}{}{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){,}\left({4}{,}{4}{,}{4}\right){=}{0}\right]\right){,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}{,}{-1}\right]\right]\right)$ (2)