Differences

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+====== Differentiate ======
+----
+====Description====
+  * ''Differentiate[var1, var2]'' differentiates expression successively with respect to ''var1'', ''var2'' ...
+  * ''Differentiate'' can differentiate tensorial expressions with respect to tensorial variables. ''Differentiate'' takes care about dummies relabelling and symmetries of tensors. The following convention is adopted:\[
+ \frac{\partial T_{m_1 \dots m_k}}{\partial T_{n_1 \dots n_k}} = \delta^{n_1}_{m_1} \times \dots \times \delta^{n_k}_{m_k} \,+\, permutations
+\]such that r.h.s. has exactly same symmetries as l.h.s.
+====Examples====
+----
+Derivative with respect to ''x'':
+<sxh groovy; gutter: false>
+println Differentiate['x'] >> 'x**n'.t
+</sxh>
+<sxh plain; gutter: false>
+   > n*x**(n-1)
+</sxh>
+----
+Derivative with respect to ''f_mn'':
+\[
+\frac{\partial }{\partial f_{mn}} \, \left(\, f_{ab} f_{cd} \, \right) = \delta_a^m \, \delta_b^n \, f_{cd} \,+\, \delta_d^n \, \delta_c^m \, f_{ab}
+\]
+Redberry code:
+<sxh groovy; gutter: false>
+println Differentiate['f_mn'] >> 'f_ab*f_cd'.t
+</sxh>
+<sxh plain; gutter: false>
+   > d_a^m*d_b^n * f_cd + d_d^n*d_c^m*f_ab
+</sxh>
+----
+Derivative involving a symbolic function ''f_mn[t_ab]'':
+\[
+\frac{\partial }{\partial t_{mn}} \, \left(\, t_{ab} f^{ab}(t_{pq}) \, \right) =  f^{mn}(t_{dc}) \,+\, t_{ab} \frac{\partial }{\partial t_{mn}} \left( f^{ab}(t_{dc})\right)
+\]
+Redberry code:
+<sxh groovy; gutter: false>
+println Differentiate['t_mn'] >> 't_ab*f^ab[t_pq]'.t
+</sxh>
+<sxh plain; gutter: false>
+   > f^{mn}[t_{dc}] + t_{ab}*f~(1)^{abmn}[t_{dc}]
+</sxh>
+----
+Derivative with respect to ''x_m'' and ''y_m'':
+\[
+\frac{\partial^4 }{\partial x_{m} \partial x^m \partial y_n \partial y^n} \, \left(\, (x_a y^a)^5 \, \right) =  240\,  (y^b x_b)^3+40\, (y^b x_b)^3 \delta^a_a + 120\, (y_m y^m) (y^b x_b) (x_a x^a)
+\]
+Redberry code:
+<sxh groovy; gutter: false>
+def diff = Differentiate['x_m', 'x^m', 'y_a', 'y^a'] & CollectScalars
+println diff >> '(x_a*y^a)**5'.t
+</sxh>
+<sxh plain; gutter: false>
+   > 240*(y^b*x_b)**3+40*(y^b*x_b)**3*d^a_a+120*y_m*y^m*y^b*x_b*x_a*x^a
+</sxh>
+----
+Derivative with respect to antisymmetric tensor:
+<sxh groovy; gutter: false>
+setAntiSymmetric 'R_pq'
+println Differentiate['R_mn'] >> 'R_ab'.t
+</sxh>
+<sxh plain; gutter: false>
+   > d_{a}^{m}*d_{b}^{n} - d_{a}^{n}*d_{b}^{m}
+</sxh>
+----
+====Options====
+''Differentiate[var1, var2, ..., transformations]'' allows to pass additional ''transformations'' which will be applied after each step of differentiation (for performance reasons):
+<sxh groovy; gutter: true>
+//setting up symmetries of Riemann tensor
+addSymmetry 'R_abcd', -[1, 0, 2, 3].p
+addSymmetry 'R_abcd', [2, 3, 0, 1].p
+def tensor = 'R^acbd*Sin[R_abcd*R^abcd]'.t
+def var1 = 'R^ma_m^b'.t,
+    var2 = 'R^mc_m^d'.t
+def diff1, diff2
+timing {
+  //take second derivative and then simplify
+  diff1 = (Differentiate[var2, var1] & ExpandAndEliminate) >> tensor
+}
+</sxh>
+<sxh plain; gutter: false>
+   > Time: 1338 ms.
+</sxh>
+<sxh groovy; gutter: true; first-line: 14>
+timing {
+  //take second derivative and simplify permanently
+  diff2 = Differentiate[var2, var1, ExpandAndEliminate] >> tensor
+}
+</sxh>
+<sxh plain; gutter: false>
+   > Time: 14 ms.
+</sxh>
+<sxh groovy; gutter: true; first-line: 18>
+assert diff1 == diff2
+</sxh>
+====See also====
+  * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:representation_of_derivatives]], [[documentation:guide:list_of_transformations]]
+  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/transformations/DifferentiateTransformation.html| DifferentiateTransformation]]
+  * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/transformations/DifferentiateTransformation.java|DifferentiateTransformation.java]]