# Differences

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 — documentation:ref:differentiate [2015/11/21 12:33] (current) Line 1: Line 1: + ====== 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'':​ + + println Differentiate['​x'​] >> '​x**n'​.t + ​ + + > n*x**(n-1) + ​ + ---- + 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: + + println Differentiate['​f_mn'​] >> '​f_ab*f_cd'​.t + ​ + + > d_a^m*d_b^n * f_cd + d_d^n*d_c^m*f_ab + ​ + + + ---- + 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: + + println Differentiate['​t_mn'​] >> '​t_ab*f^ab[t_pq]'​.t + ​ + + > f^{mn}[t_{dc}] + t_{ab}*f~(1)^{abmn}[t_{dc}] + ​ + + ---- + 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: + + def diff = Differentiate['​x_m',​ '​x^m',​ '​y_a',​ '​y^a'​] & CollectScalars + println diff >> '​(x_a*y^a)**5'​.t + ​ + + > 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 + ​ + + ---- + Derivative with respect to antisymmetric tensor: + + setAntiSymmetric '​R_pq'​ + println Differentiate['​R_mn'​] >> '​R_ab'​.t + ​ + + > d_{a}^{m}*d_{b}^{n} - d_{a}^{n}*d_{b}^{m} + ​ + ---- + + ====Options==== + ''​Differentiate[var1,​ var2, ..., transformations]''​ allows to pass additional ''​transformations''​ which will be applied after each step of differentiation (for performance reasons): + + //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 ​ + } + ​ + + > Time: 1338 ms. + ​ + + timing { + //take second derivative and simplify permanently + diff2 = Differentiate[var2,​ var1, ExpandAndEliminate] >> tensor + } + ​ + + > Time: 14 ms. + ​ + + assert diff1 == diff2 + ​ + + ====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]]