Advanced Calculus using Mathematica

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

Mathematical Themes

The association between the various cases of nonlinear equations and their linear tangent equations is the fundamental idea of differential calculus: smooth functions are “locally linear.”   More graphically speaking, a sufficiently magnified view of a smooth nonlinear graph appears to be the same as the graph of the linear tangent equation.  To understand this idea, we treat the geometry of  equations in the explicit, implicit, and parametric cases by increasing dimension.  Each chapter begins with the linear case.  

Explicit, implicit, and parametric representations

Students learning calculus of several variables face a bewildering number of formulas if they try to learn the subject as a special formula for every case.  “The gradient is perpendicular to the tangent” is false for explicit surfaces, z=f[x,y], but true for implicit surfaces, F[x,y,z]=c.  It seems there are many equations for tangents.  

Unified procedures that apply to explicit, implicit, and parametric cases

However, if we are conscious of the explicit, implicit, parametric classification of equations and graphs, there is only one procedure for finding the equation of the tangent.  The unified procedure reduces the nonlinear problem to a linear one that locally approximates the original.

In the case of an explicit nonlinear equation and graph, the tangent equation procedure produces an explicit linear equation and graph.  The tangent procedure applied to an implicit nonlinear equation (or contour) and graph produces an implicit linear tangent equation and graph.  The case of a nonlinear parametric equation and graph produces a linear parametric tangent equation and graph.  This works in various dimensions for curves and surfaces.  

In every case the same 3 steps produce the equation of the tangent: (1) compute the symbolic total differential. (2) evaluate the partial derivatives at the point of tangency. (3) change local coordinates back to x-y-z or plot in local coordinates.   Various cases are as follows:

Explicit 3D: Procedure 3.3.1

Implicit 2D: Procedure 4.2.1

Implicit 3D: Procedure 4.5.1

Parametric 1D: Procedure 9.6.1

Parametric 2D: Procedure 15.3.1

The links above give specific analytical examples of each case. Geometrically, the computations give the result of "zooming in" as in the next section.

Local linearity: microscopic analysis of smooth quantities

A geometric interpretation of the tangent procedure is that the procedure computes what we would see in an infinitely (or “sufficiently”) powerful microscope.

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:

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.

Local linearity & Integration

The idea of many integral formulas is to sum the “microscopic amounts” and show that micrsocopic errors don’t accumulate.  (The idea that errors that are small in the microscope don’t accumulate is called Duhamel’s Principle.)  An important basic case of this idea arises in the Fundamental Theorem of Calculus in 1 variable.  That example shows how a uniform approximation of the derivative (smoothness or ) is the needed technical condition.  Other examples of sums of microscopic amounts animated in the text are:

Arclength: Theorem 9.7.1

Work or Flux: 2D

Green’s Theorem: 2D

Surface Area: 3D

Stokes’ and Gauss’Theorems