Table of Contents

Representation of derivatives

Next topic: Applying and manipulating transformations

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 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