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--- documentation:ref:tensorfield [2015/11/21 12:33]
+++ documentation:ref:tensorfield [2015/11/21 12:33] (current)
@@ Line 1: / Line 1: @@
+====== TensorField ======
+----
+====Basics====
+''TensorField'' represents a function like ''f_mn[x_a]'' or derivative of function like ''f~(1)_a[x_i]''. Like any other expression, it inherits all properties of [[Tensor]]. Additionally it inherits all properties of [[SimpleTensor]] and adopts same convention on its indices: the indices of ''TensorField'' ([[SimpleIndices]]) are sorted by their [[documentation:ref:IndexType|types]], but the relative ordering of indices within a particular type is same as defined by the user:
+<sxh groovy; gutter:false>
+def t = 'f_cba^\\alpha_AB[t_mn, f_i]'.t
+println t.indices
+</sxh>
+<sxh plain; gutter:false>
+    > _{cbaAB}^{\alpha}
+</sxh>
+The relative ordering is not changed even if tensor has symmetry property; in latter case Redberry uses algorithms provided by [[documentation:guide:Mappings of indices]] to compare expressions instead of trying to put expression in canonical form.
+[[documentation:guide:symmetries_of_tensors|Symmetries]] of ''TensorField'' can be specified exactly in the same way as for [[SimpleTensor]]:
+<sxh groovy; gutter: false>
+addSymmetry 'f_pqr[k_l, k_l]', [1, 0].p
+//or equivalently
+'f_klm[k_a, k_b]'.t.indices.symmetries.addSymmetry( [1, 0].p )
+</sxh>
+<html><div><p style="padding-bottom:15px">   </p></div></html>
+In case of derivatives, the total indices of ''TensorField'' are formed by concatenation of initial indices and inverted indices of differentiating variables. For example, the indices of
+\[
+ \frac{\delta}{\delta f^{abc}} \frac{\delta}{\delta f^{de}} F_{mn}\left(f_{abc}, f_{ab}\right)
+\]
+will be formed by sequential concatenation of initial indices $_{mn}$, indices of differentiating variable corresponding to first argument $_{abc}$ and  indices of differentiating variable corresponding to second argument $_{de}$; thus, the resulting indices will be $_{mnabcde}$ and the corresponding tensor field
+<sxh groovy; gutter: false>
+'F~(1, 1)_{mn abc de}[f_abc, f_ab]'.t
+</sxh>
+Additionally, if derivative taken several times with respect to same slot, the corresponding indices will be symmetric, since the relative order of differentiation is not relevant:
+<sxh groovy; gutter: false>
+println 'F~(2)_{mn ab cd}[f_ab] + F~(2)_{mn cd ab}[f_ab]'.t
+</sxh>
+<sxh plain; gutter: false>
+   > F~(2)_{mn ab cd}[f_ab]
+</sxh>
+<html><div><p style="padding-bottom:15px">   </p></div></html>
+Since ''TensorField'' inherits both [[Tensor]] and [[SimpleTensor]] it provides all features found in them like container properties, utterable, symmetries etc. On the other hand, the ''.name'' property from [[SimpleTensor]] will additionally encode information about arguments of ''TensorFiels'', so e.g. functions like ''f_i[x_a]'' and ''f_i[x_ab]'' will have different ''.name''.
+====Additional features====
+The additional features of ''TensorField'' appear when it represents derivative:
+<html>
+<table style="width:600px;" cellpadding="10">
+<tr>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.isDerivative()</code></td>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns whether <code>TensorField</code> represents derivative of function (e.g. <code>f~(3)[x]</code>) or just function (e.g. <code>f[x]</code>)</td>
+</tr>
+<tr>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.getDerivativeOrder(i)</code></td>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns order of derivative with respect to argument at <i>i</i>-th position</td>
+</tr>
+<tr>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.parentField</code></td>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns function from which the derivative is taken </td>
+</tr>
+<tr>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.partitionOfIndices</code></td>
+  <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns partition of derivative indices; this partition separates indices of parent function and indices appended by derivatives </td>
+</tr>
+</table>
+</html>
+Let us consider for example derivative
+\(\displaystyle
+ \frac{\delta}{\delta x^{cd}} \frac{\delta^2}{\delta t^m \, \delta t^n}  F_{ab}\left(x_{mn}, t_i, x\right)
+\):
+<sxh groovy; gutter: true>
+def f = 'F~(1,2,0)_{ab cd mn}[x_mn, t_i, x]'.t
+assert f.isDerivative()
+println f.getDeivativeOrder(1)
+</sxh>
+<sxh plain; gutter: false>
+   > 2
+</sxh>
+<sxh groovy; gutter: true; first-line: 4>
+println f.parentField
+</sxh>
+<sxh plain; gutter: false>
+   > F_{ab}[x_{mn},t_{i},x]
+</sxh>
+<sxh groovy; gutter: true; first-line: 5>
+println( (0..(f.size() - 1)).collect { f.getDeivativeOrder(it) })
+</sxh>
+<sxh plain; gutter: false>
+   > [1, 2, 0]
+</sxh>
+<sxh groovy; gutter: true; first-line: 6>
+println f.partitionOfIndices
+</sxh>
+<sxh plain; gutter: false>
+   > [[_{ab}], [_{cd}], [_{m}, _{n}], []]
+</sxh>
+The first element in ''.partitionOfIndices'' is indices of parent field, while //(i+1)//-th element of this array corresponds to indices of derivatives with respect to //i//-th slot.
+====See also====
+  * Related guides: [[documentation:guide:tensors_and_indices]], [[documentation:guide:representation_of_derivatives]], [[documentation:guide:symmetries_of_tensors]]
+  * Reference: [[documentation:ref:tensor]], [[documentation:ref:simpletensor]], [[documentation:ref:simpleindices]], [[documentation:ref:differentiate]]
+  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/tensor/TensorField.html| TensorField]]
+  * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/tensor/TensorField.java|TensorField.java]]