Singular Manual: diff

  | ### 5.1.24 diff `Syntax:` `diff (` poly_expression`,` ring_variable `)` `diff (` vector_expression`,` ring_variable `)` `diff (` ideal_expression`,` ring_variable `)` `diff (` module_expression`,` ring_variable `)` `diff (` matrix_expression`,` ring_variable `)` `Type:` the same as the type of the first argument `Syntax:` `diff (` ideal_expression`,` ideal_expression `)` `Type:` matrix `Syntax:` `diff (` number_expression`,` ring_parameter `)` `Type:` number `Purpose:` computes the partial derivative of a polynomial object by a ring variable (first forms) respectively differentiates each polynomial (1..n) of the second ideal by the differential operator corresponding to each polynomial (1..m) in the first ideal, producing an m x n matrix. respectively if the coefficient ring is a transcendental field extension, differentiates a number (that is, a rational function) by a transcendental variable (ring parameter). `Example:`
        ring r=0,(x,y,z),dp;
        poly f=2x3y+3z5;
        diff(f,x);
      ==> 6x2y
        vector v=[f,y2+z];
        diff(v,z);
      ==> 15z4*gen(1)+gen(2)
        ideal j=x2-yz,xyz;
        ideal i=x2,x2+yz,xyz;
        // corresponds to differential operators
        // d2/dx2, d2/dx2+d2/dydz, d3/dxdydz:
        print(diff(i,j));
      ==> 2,0,
      ==> 1,x,
      ==> 0,1
        // differentiation of rational functions:
        ring R=(0,t),(x),dp;
        number f = t^2/(1-t)^2;
        diff(f,t);
      ==> (-2t)/(t3-3t2+3t-1)
See contract; ideal; jacob; matrix; module; poly; var; vector.

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