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 — documentation:guide:representation_of_derivatives [2015/11/21 12:33] (current) Line 1: Line 1: + ====== Representation of derivatives ====== + <​html>​ +
+ ​ + Next topic: [[documentation:​guide:​applying_and_manipulating_transformations]] + <​html>​ + ​ + ​ + ​ + + ---- + Representation ​ of derivatives in Redberry is very similar to other systems and + is very close to standard mathematical sense of this concept. However, presence + of indices brings new features of derivatives with respect to indexed arguments. ​ + + ====Indexless expressions==== + Consider first the common mathematical notation for derivatives of ordinary ​ + functions. The standard notation $f'​(y)$ is really a shorthand for $\left. ​ + \frac{d}{d x} f(x)\right|_{x = y}$ etc. Redberry uses the ''​%%~%%''​ + symbol followed by the derivative order (or orders in case of function with + several arguments) instead of primes. So, for example, the following expression + + def t = '​f~(3)[x**2]'​.t + ​ + represents $\left. \frac{d^3}{d t^3} f\left(t\right) \right|_{t = x^2}$. When substituting e.g. $f(t) = \sin t$ in the above expression one will have + + println 'f[x] = Sin[x]'​.t >> '​f~(3)[x**2]'​.t + ​ + + > -Cos[x**2] + ​ + + In the case of multivariate functions, one should specify how many times to + differentiate with respect to each slot (argument): + + def t = 'f~(2, 3, 0)[a**2, w, q]'.t + ​ + represents $\displaystyle \left. \frac{\partial^5}{\partial x^2 \partial y^3} + f\left(x, y, q\right) \right|_{x = a^2 ,\, y =w}$. The above notation applies to derivatives of pure tensor fields; if one need to take derivative of some particular expression one should use [[documentation:​ref:​differentiate|Differentiate]] ​ transformation or use ''​D[vars][exp]''​ syntax: + + println 'D[x, y][y*x**2 + x*y**2 + f[x, y]]'.t + ​ + + > 2*x + 2*y + f~(1,1)[x, y] + ​ + + + ====Indexed expressions==== + In the case of indexed objects one should append additional indices of differentiation variables. For example, expression + + def t = '​F~(2)_{mn ab}^{cd}[f_ab]'​.t + ​ + represents $\left. \frac{\delta}{\delta t^{ab}} ​ + \frac{\delta}{\delta t_{cd}} F_{mn}\left(t_{ab}\right) \right|_{t_{ab} = f_{ab}} ​ +$. As we see, the inverted indices of differentiation variables should be appended ​ + to the indices of pure tensor field. Since the relative ordering of derivatives is + irrelevant, the indices of derivative have additional symmetries. From the previous ​ + example: + + println t.indices.symmetries.permutationGroup + ​ + + > Group( +[[2, 4], [3, 5]] ) + ​ + + + Indices of differentiation variables are appended sequentially starting from the first argument. So, for example, expression: + + def t = 'F~(1, 1)_{mnabcde}[f_abc,​ f_ab]'​.t + ​ + represents $\displaystyle \frac{\delta}{\delta f^{abc}} \frac{\delta}{\delta ​ + f^{de}} F_{mn}\left(f_{abc},​ f_{ab}\right)$ but not $\displaystyle ​ + \frac{\delta}{\delta f^{cde}} \frac{\delta}{\delta f^{ab}} F_{mn}\left(f_{abc}, ​ + f_{ab}\right)$. + ====See also==== + * Related reference material: [[documentation:​ref:​differentiate]]