Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.

Analytical Local Linearity

The analytical explanation of zooming to the linear spproximation in the case of an explicit curve or surface is based directly on the “small oh” formula:

The graph of z=f[x,y] has a tangent at (x,y) if ε→0 as the change d X=(d x,d y)→0.

If x and y are fixed and you focus your microscope at the point (x,y,f[x,y]), the linear term in the change (d x,d y) is given by m·d x+n·d y with fixed values and ,

f[x+d x,y+d y]-f[x,y]=m·d x+n·d y+ε· |d X|

  The difference between the nonlinear and linear terms for fixed (x,y) and a change of d X=(d x,d y) is

Nonlinear Change - Linear Change=(f[x+d x,y+d y]-f[x,y])-(m·d x+n·d y)=ε· |d X|

The tangency condition above means that if the size of the change is sufficiently small, |d X|≈0 then ε≈0.  Magnification by M means we graph for changes |d X|<1/M and then

Magnified Difference in Change=|Nonlinear Change - Linear Change|·M<ε

We can not see the difference between the linear and nonlinear change once the magnification is sufficient to make the error ε below the “resolution” of our microscope.  

In a first course on multivariable calculus, most students struggle to understand the "limit"  ε→0 as the change d X=(d x,d y)→0 (2D limits are hard!), but the conceptually equivalent "microscopic approximation" can still give students a clear idea of how to use partial derivatives.  (The limit should also be "uniform" to make the ideas work in the implicit function theorem and integration.  See technical smoothness below.)

Extra conditions for implicit and parametric graphs are explained in the text.  Local linearity and “zooming” also applies to vector fields and flows as described in the novel topics below.

Technical smoothness

Smoothness can be explained intuitively by the question: What happens if we move the focus point of our infinitesimal microscope a small amount?  In 1 variable we want the line we see to only change slope by a small amount.  This is continuity of the derivative, but comes directly from the limiting slope approximation if we take a “strong” notion of limit.

Smooth or can be defined directly as a locally uniform limit or Peano’s double limit in 1 variable, but since most beginning computations are based on the complex analytic classical functions (sine, cosine, log, exponential), students can check smoothness by computation (complex derivatives are automatically uniform and given by the same rules).  This is easy to check, but important for example in correct use of Green’s Theorem.

The 2D curl of the winding vestor field is zero (except at the origin), so the double integral in Green’s theorem is zero.  However, the path integral around the unit circle is 2π and it seems that either 0=2π or Green’s formula is false.  The resolution of this dilemma is that the winding field is not smooth on the whole interior of the circle, so Green’s Theorem does not apply.

The difference between the intuitive microscopic view and technical pointwise defined partial derivatives is given in the optional Section 3.9.  We feel these negative examples should not be emphasized for beginning students, but they are certainly interesting for advanced students (like our graduate TAs) and Mathematica lets us “see” things like continuous nowhere differentiable functions.  The uniform theory is “smooth” and intuitive, while the pointwise counterexamples show why a strong limit is needed to make many of the formulas of calculus true.