Multivariable Differentiability Applet

This applet follows the same approach as the multivariable continuity applet, trying to build intuition the definition of differentiability for a function of two variables.

• The definition of continuity can be understood in terms of boxes and approximations of constant functions.  If a function is continuous, for any given error height, we can find a width where the function is within a box.  (To understand the definition we draw a rectangle in the one variable case and a hat box in the two variable case.  In either case this is a constant function plus or minus the error.)
• The definition of differentiability can be understood in terms of cones and linear functions (planes).  For functions of one variable, differentiability says that for any given error in slopes of the tangent line, we can find a width where the function is within a cone of two lines.  (The cone is the candidate tangent line with the slopes wiggled up and down by the error tolerance.)
• For functions of two variables, wiggling the tangent plane produces a cone.  If a function is differentiable, for any given error in slopes of the tangent plane, we can find a width where the function is within the cone.

As with continuity, to prove differentiability at a point we would need to prove a rule that gives a delta for any positive epsilon, with the function trapped by the approximating cone.  As with continuity, we will probably be convinced if we can find the deltas that go along with the epsilon values of .1, .01, and .001.

Click on the button below to launch the applet.
• The applet has a control window and 3 graphics windows.
• The first graphical window is the domain window.  It contains shows a grid of points, a big red dot (the hot spot with coordinates (C_1,C_2)), a green ring or radius ε centered at the hotspot, and a yellow dot at the edge of the disk.  These two dots are hot spots and can moved with a mouse click and drag.
• The values of the coordinates are on the control panel.  The values of ε and δ can either be typed in or shifted with the arrow buttons.
• The graph window shows a graph of the function, with a red dot at the graph of the hot spot and a green patch for the graph of the green circle of radius &epsilon.  It also has two halves of the approximating cone above and below the graph of the hot spot.   With checkboxes on the control panel you can remove the cone or add in a small section of the candidate tangent plane.
• The close up window zooms in on the disks.  For a given ε, you want to find a δ where the green surface is trapped between the two halves of the pink cone.

Things to notice:
• Sometimes it is hard to understand differentiability since almost all functions we can think of are differentiable almost everywhere.  Consider the function f(x,y) = 2*x*y/sqrt(x^2+y^2).  (This function in polar coordinates translates to r*sin(2*theta).)  It is differentiable everywhere except at the origin.  Compare what happens at the origin with what happens away from the origin.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.