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The idea of the chain rule
An illustration of the basic concept of the chain rule using interactive graphics to diagram the relevant points on the graphs and the corresponding slopes.
Multivariable chain rule examples
Examples demonstrating the chain rule for multivariable functions.
Introduction to the multivariable chain rule
Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.
Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case.
A refresher on the chain rule
How to compute the derivative of a composition of functions.
Simple examples of using the chain rule
Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions.
Definition: Chaos definition
A dynamical system exhibits chaos if it has solutions that appear to be quite random and the solutions exhibit sensitive dependence on initial conditions.
Characterization of linear 2D flows
A model of chemical pollution in a lake
A discrete dynamical system model of chemical pollution in a lake illustrates the concept of movement toward equilibrium.
Chemical pollution model exercises answers
Answers to selected Exercises on solving dynamical systems for chemical pollution in a lake.
Chemical pollution model exercises
Exercises on solving dynamical systems for chemical pollution in a lake to determine the movement toward equilibrium.
Calculating the formula for circulation per unit area
A sketch of the proof for the formula for the component of the curl of a vector field.
Classic examples of Taylor polynomials
Some of the most famous (and important) examples of Taylor expansions.
Definition: Closed curve definition
A closed curve is a curve where the beginning and end points are the same.
Definition: Clustering coefficient definition
The clustering coefficient is a measure of the number of triangles in a graph.
Using cobwebbing as a graphical solution technique for discrete dynamical systems
How to use cobwebbing to approximate the solution of discrete dynamical systems.
Definition: Codomain definition
The codomain of a function is the set of its possible outputs.
Definition: Composition definition
The composition of two functions is the function formed by applying the original two functions in succession.
Computational tricks regarding Taylor polynomials
A short cut to calculating Taylor polynomials in terms of Taylor polynomials of simple functions.
Connecting network structure to dynamical properties
An exploration of the influence of network structure on the properties of dynamical processes taking place on networks.
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