Subject: Maxima by Example, Ch. 6, Differential Calculus
From: Edwin Woollett
Date: Sat, 19 Jul 2008 14:02:26 -0700
Maxima by Example, Chap. 6, Differential Calculus, files have
been posted on my webpage: http://www.csulb.edu/~woollett .
The files available are mbe6calc1.pdf, vcalc.mac, vcalcdem.mac (meant to
be used via batch(vcalcdem) ), calc1code.txt, cylinder.mac,
and sphere.mac (the latter two also meant to be batched in).
The organization of chapter six is:
6. Differential Calculus
6.1 Differentiation of Explicit Functions
6.1.1 diff
6.1.2 Total Differential
6.1.3 Controlling the Form of a Derivative
with gradef(..)
6.2 Critical and Inflection Points of a
Curve Defined by an Explicit Function
6.2.1 Example 1: A Polynomial
6.2.2 Automating Derivative Plots with
plotderiv(..)
6.2.3 Example 2: Another Polynomial
6.2.4 Example 3: x^(2/3), Singular Derivative,
Use of limit(..)
6.3 Tangent and Normal of a Point of
a Curve Defined by an Explicit Function
6.3.1 Example 1: x^2
6.3.2 Example 2: ln(x)
6.4 Maxima and Minima of a Function
of Two Variables
6.4.1 Example 1: Minimize the Area of a
Rectangular Box of Fixed Volume
6.4.2 Example 2: Maximize the Cross
Sectional Area of a Trough
6.5 Tangent and Normal of a Point of
a Curve Defined by an Implicit Function
6.5.1 Tangent of a Point of a Curve Defined
by f(x,y) = 0
6.5.2 Example 1: Tangent and Normal of a
Point of a Circle
6.5.3 Example 2: Tangent and Normal of a
Point of the Curve sin(2*x)*cos(y) = 0.5
6.5.4 Example 4: Tangent and Normal of a
Point of the Curve: x = sin(t), y = sin(2*t)
6.5.5 Example 5: Polar Plot:
x = r(t)*cos(t), y = r(t)*sin(t)
6.6 Limit Examples Using Maxima's
limit(..) Function
6.6.1 Discontinuous Functions
6.6.2 Indefinite Limits
6.7 Taylor Series Expansions using taylor(..)
6.8 Vector Calculus Calculations and
Derivations using vcalc.mac
6.9 Maxima Derivation of Vector Calculus
Formulas in Cylindrical Coordinates
6.9.1 The Calculus Chain Rule in Maxima
6.9.2 Laplacian del^2 f(rho,phi,z)
6.9.3 Gradient del f(rho,phi,z)
6.9.4 Divergence del dot B(rho,phi,z)
6.9.5 Curl del cross B(rho,phi,z)
6.10 Maxima Derivation of Vector Calculus
Formulas in Spherical Polar Coordinates
Some comments about use of the package vcalc.mac
to calculate the gradient, divergence, curl, and Laplacian
(of either a scalar or a 3-vector) in cartesian, cylindrical,
and spherical polar coordinate systems.
The batch file vcalcdem.mac takes the reader through
examples of use.
An example of using this package would be "curl([r*cos(theta),0,0]);"
(if the current coordinate system has already been changed to spherical
polar) or "curl([r*cos(theta),0,0], s(r,theta,phi) );" if the
coordinate system needs to be shifted to spherical polar from
either cartesian (the starting default) or cylindrical.
The order of list vector components corresponds to the order of the
arguments in "s(r,theta,phi)".
The Maxima output is the list of the vector curl components in the
current coordinate system, in this case "[0, 0, sin(theta)]" plus
a reminder to the user of what the current coordinate system is
and what symbols are currently being used for the independent
variables.
Thus the syntax is based on lists and is similar to (although better
than ! ) Mathematica's syntax.
There is a separate function to change the current coordinate system.
An example of its use would be:
"setcoord( cy(rho,phi,z) );" to set the use of cylindrical
coordinates (rho,phi,z), or "setcoord( cy(r,t,z) );" to set
cylindrical coordinates (r,t,z).
The package vcalc.mac also contains the plotting function
plotderiv(..) which is useful for "automating" the plotting
of a function and its first n derivatives, and the function
lcross(..) which can be used to find the cross product of 3-vectors
represented by lists.