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)); |
> |