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.
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}
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