Line functions and polynomials, exponential and trig functions, and
rational functions, and how they all grow. Interpolation of data sets with
curves. Best fit lines through data sets. Limiting behavior of functions
as variables grow and grow. Average Growth and growth rates and
percentage growth rates. Instantaneous Growth rates and
instantaneous percentage growth rates. The derivative as instantaneous
growth rate. What measurements the derivative facilitates. Optimization
(Max and Min of functions of one and even more variables).
The derivative measures instantaneous growth rate. The chain rule, product
rule, and their children. Practice at hand calculation of derivatives.
What it means for f'[x] to be positive, negative and zero. Three standard
differential equations and what the differential equations alone tell you about
their solutions. Comparing logistic and exponential growth.
The Race Track Principle says that horses running faster beat horses
running slower. It's sometimes known as the active voice of the Mean
Value Theorem, and is the most used form of that theorem in calculus.
Euler's method for approximating functions given their derivatives, and how
that is used in studying differential equations. Further experience with
differential equations with a big dose of qualitative examination of the
models. Models studied are linear, exponential, logistic, and versions of
the predator-prey system.
The integral is a definite integral, and measures area. Fundamental
integrals based on that definition. Approximation of integrals by
trapezoids. The fundamental formula of calculus says that if you know how
a function changes (its derivative) and where it starts, then you know the
function. Building a list of integrals you can calculate by hand.
Measurements based of slicing: Area, Volume, Mass and Centroids. Using
the chain rule together with the fundamental formula to help calculate
integrals by simplifying them. An excursion into bell shaped curves and
normal probability distribution.
Understanding the plot of z=f[x,y], and using slicing and accumulation to
measure volumes the surface traps. The Gauss-Green Formula, and how it
can help in such calculations. Separating variables and integrating to
solve some differential equations. Integration by parts. Complex numbers
and Euler's formula for eit. Using undetermined coefficients
to find formulas for certain integrals. Various special substitutions for
simplifying calculation of integrals.
Splines for making smooth transitions between different types of
functions. Smoothness of transition as "order of contact" to indicate
better and better approximation of functions. Taylor polynomials as
polynomials having higher and higher order of contact with the function
you started with. The standard power series: Geometric, Exponential, Trig
and ArcTan. Newton's method via order of contact principles. Using
expansions to calculate limits at finite points and L'Hospital's rule.
Taylor's formula from order of contact. Order of contact from series
expansions. Barriers to convergence, and complex singularities.
Convergence intervals via barriers, and from the root and ratio tests.
Functions defined by power series, and derivation of things like Euler's
formula. Power series for solving certain differential equations.
Convergence intervals for those solutions. Convergence intervals simply as
determined by the behavior of an xn.
Parametric representation of some curves and surfaces. Do parametric
curves cross? If so, do they collide? Represent 2D and 3D vectors: Basic
geometry and algebra of vectors. Tangent vectors, normal vectors, tangent
lines. Dot products and perpendicularity. Cross products and areas.
Lines and planes in 3 dimensions.
Using vectors to represent line functions, planes. Parametric curves as
flows. Arc length. Plotting vector functions in 2D and 3D. Measuring
curvature.
The gradient and the chain rule. The gradient points in the direction of
greatest initial increase. Linearization: The chain rule and gradient rule
in linearization. Level curves and surfaces. Optimization. Lagrange
multipliers for constrained optimization. Vector fields associated with
fluid flow. Solutions of y'[t] = f[t,y[t]] as trajectories in a vector
field. Measuring flow along and across curves in the plane. Path
independence and gradient fields. Recognizing gradient fields.
Divergence and rotation of 2D fields. Using the Gauss-Green formula to
measure flows. Singularity sources and sinks, and swirls. Changing
coordinate systems to simplify integrals. Linearizing coordinate changes
to find the fudge factor. Using the Jacobian in calculating integrals.
Using it all to calculate divergence and rotation. The
Laplacian, and the impossibility of interior maxima for harmonic functions.
Some special 3D coordinate systems, and learning to use them: Cylindrical
and Spherical, especially. Using variables changes and the Jacobian in 3D
to simplify calculations. Measuring flows across surfaces. The divergence
theorem as the 3D Gauss-Green theorem. Measuring flow around boundaries
of surface patches. Stokes' theorem as 3D Gauss-Green theorem, and the
curl is 3D rotation.
A brief review of the solutions of the exponential differential equation
starts us off. This leads to
the object: a study of the second order equation
y'' + b y' + c y = F.
Experience is gained with understanding the roles of the coefficients b
and c in the homogeneous equation. Initial value problems are examined
numerically and formulas are derived. Superposition principle. The
convolution integral method as a reliable solution scheme. Step functions
and impulse function responses. Beats and resonance. Guessing solution
forms in simple situations. Using trig approximations and introductory
Fourier Series solutions. Introductory Laplace transform methods.
Examine the exponential and logistic equations, and modifications of them,
again. Emphasize the relation between the vector fields and the
trajectories. Examine phase lines and bifurcation plots: Get the
qualitative information you can from the equation. Begin a study of first
order systems with lots of visual experience with 2D linear systems, and
with examination of various modifications of predator-prey models. Look
at critical points and phase plots and their relations to the vector
fields associated. Use information from 2D linear systems to predict
qualitative behavior for Van der Pol's oscillator.
Examine 2D and 3D linear, constant coefficient systems through geometric
and analytic means. Emphasize the role of the eigensystems of the
coefficient matrices: What qualitative information do they give? How do
you see the eigensystems in phase plots? How do they lead to nice
formulas? How does that relate to oscillator equations and other constant
coefficient higher order equations?
Revisit the nonlinear models from before: Predator-Prey and modifications,
Van der Pol. Throw in the pendulum equation. Examine qualitative
behavior further. Look at local linearization of the systems to find
features of the flows. A brief look at chaos and what might cause it in
the Lorenz oscillator. Gradient systems and Hamiltonian systems.
Quickly review basic geometry of vectors in 2D and 3D. Hanging geometric
objects on perpendicular frames in 2D and 3D. Matrix multiplication.
What does hitting a circle with a 2D matrix do? Stretching and rotating
with special 2D matrices. Using perpendicular frame matrices and
stretching matrices to build special matrices. Using the Singular Value
Decomposition to see that you have done it all. Inverting matrices and the
SVD. Solving equations and the SVD. Rank, Nullity and Determinant via
SVD. Introductory hand calculations using Cramer's rule and Gaussian
elimination.
Carry on with the SVD in 3D. All matrices are still constructed from pairs
of perp frame matrices and stretcher matrices. All calculations of rank
and nullity carry over. Recognizing when a given system of n linear
equations in k unknowns has a solution, or many solutions, or none.
Finding solutions when they exist. Constructing Pseudo Inverses. Roundoff
of SVD matrices and some image compression. Principal Component
Analysis. Ill conditioned matrices, and using the SVD to recognize them.
Projections onto subspaces. Calculating the dimension of subspaces.
Linear dependence, orthonormals sets, and Gram Schmidt process.
Eigenvalues,eigenvectors and using them to recognize diagonalizable
matrices. Complex eigenvalues and eigenvectors.The matrix exponential for
diagonalizable and non-diagonalizable matrices. Using matrix exponentials
and matrix powers to solve continuous dynamical systems (systems of linear
diffeerential equations) and discrete dynamical systems (systems of
difference equations). Discussion of the spectral theorem and its proof.
Given an arbitrary matrix A, using the spectral theorem applied to
Transpose[A].A to explain why every matrix has a singular value
decomposition. Positive definite and positive semidefinite matrices.
Quadratic forms. Grammian matrices. Functions as vectors. The dot
product of two functions. The component of one function in the direction
of another. Orthogonal sets of functions: Sine systems, Cos
systems, Sin-Cosine systems, Legendre Polynomial system. Sets of functions
orthogonal with respect to a weight function. Chebyshev polynomials.
Gram-Schmidt process. Fourier approximation and orthogonal functions.
