A Non-Differentiable Surface
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This animation depicts a surface sometimes studied in multivariable calculus.
(See notes below.)
The surface shown here is given by the equation
z = x3 ⁄ (x2 + y2) is also continuous
at the origin (we put z = 0 at the origin) and both of the
first order partial derivatives are defined there. (Compare this surface with
the other
non-differentiable surface.)
In fact, this surface has
directional derivatives in every direction at the origin.
(That is, each of the curves in which
the surface intersects a plane containing the z-axis has a tangent line at
the origin. In order to see this, let y = m x. Then z = x ⁄ (1 + m2),
so that the surface meets the plane y = m x in a straight line.) Nevertheless,
the function is not differentiable there. (The surface has no tangent plane
at the origin.) (07/27/07)