An Applet for Multivariable Visual Chain Rule

This applet looks at the chain rule in the case when a parameterized curve from R to R2 is composed with a function from R2 to R to produce a function from R to R.   The standard situation starts with a curve in the x-y plane parameterized by t, then considers z as a function of x and y. In the composition we then consider z a as a function of t.

In understanding this version of the chain rule we want to introduce a second parameter, s, so that we think of x as a function of s and y as a function of t.  The parameterized curve corresponds to the diagonal line in the s-t plane

You may find it useful to review the chain rule for functions of one variable.

The formula for the chain rule is
The applet has 4 windows, a control panel,  a Zxy window that plots z as a function of x and y, a Zst window that plots z as a function of s and t, and an "Edge and Diagonal Paths" window that shows curves together.

A Worksheet for the Chain Rule Applet for functions of two variables is available.

Things to notice:
• The Zxy graph is centered around the point P0=(x0 = x(t0), y0 = y(t0), z0 = z(x(t0)), y(t0))).   The other graphs are similarly centered around points connected to t0.
• The variable del in the control window is a zoom feature.  If you bring del down to 0.001, you will have zoomed in enough that for functions typically used in calculus classes, all of the curves are indistinguishable from their tangent lines and all surfaces are indistinguishable from their tangent planes.
• The chain rule expresses the derivative as a sum of products of derivatives.
• The sum part of the rule intuitively says that the change in z along the yellow diagonal path is the sum of the changes in z along the blue and pink paths.
• The product parts of the rule are each cases of  the single variable chain rule.  For example, the blue path plots z(x(s),y0) where z is a function of x which is a function of s.  Similarly, the pink path graphs z(x1,y(t)).
• The  readouts on the control panel give numeric information with variable names strung together to reduce the number of readout boxes.  Thus x0y0z0 gives (x0, y0, z0) and dzdxdxdt gives (dz/dx, dx/dt).
• The values in the readout  boxes can be used to verify that the formula for the chain rule works at any given value of t0.

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.