Math 13 Fall 2004

Calculus of Vector-valued Functions

Example of a function that has

different mixed partial derivatives at (0,0)

October 4, 2004

   Define a scalar-valued function of two variables

>	f := (x, y) -> x * y * (x^2 - y^2) / (x^2 + y^2);

    Have a look at its graph

>	plot3d(f(x, y), x = -1..1, y = -1..1);

      Both partial derivatives of f are continuous everywhere,

      so f is differentiable at (0, 0)

>	f_x := factor(diff(f(x, y), x)); f_y := factor(diff(f(x, y), y));

>	plot3d(f_x, x = -2..2, y = -2..2); plot3d(f_y, x = -2..2, y = -2..2);

     Let's explicitly compute the mixed partial derivatives of f at (0, 0)

>	f_xy_00 := limit(subs(x = h, y = 0, f_x) / h, h = 0); f_yx_00 := limit(subs(x = 0, y = h, f_x) / h, h = 0);

    They are different!!!

    Let's plot both mixed partial derivatives of f

>	plot3d(diff(f(x, y), y, x), x = -1..1, y = -1..1); plot3d(diff(f(x, y), x, y), x = -1..1, y = -1..1);

   They are obviously discontinuous!!!

     Remark: mixed partial derivatives are the same away from (0, 0)

>	factor(diff(f(x, y), y, x)); factor(diff(f(x, y), x, y));

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