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- Determining stability by cobwebbing linear approximations around equilibria
By cobwebbing the linear approximation to a discrete dynamical around an equilibrium, one can determine a criterion for the stability of the equilibrium. - Determining tolerance/error in Taylor polynomials.
Calculating with what tolerance a Taylor polynomial approximates a function on an interval. - Definition: Difference form definition
A discrete dynamical system is in difference form when it is written so that the left hand side gives the difference in the state variable over one time step. - The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable. - Introduction to differentiability in higher dimensions
An introduction to the basic concept of the differentiability of a function of multiple variables. Discussion centers around the existence of a tangent plane to a function of two variables. - Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable. - A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable. - Definition: Directed graph definition
A directed graph is graph, i.e., a set of objects (called vertices or nodes) that are connected together, where all the edges are directed from one vertex to another. - Derivation of the directional derivative and the gradient
Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions. - Directional derivative and gradient examples
Examples of calculating the directional derivative and the gradient. - An introduction to the directional derivative and the gradient
Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables. - Definition: Discrete dynamical system definition
A discrete dynamical system is a dynamical system whose state evolves over state space in discrete time steps according to a fixed rule. - Solutions to elementary discrete dynamical systems biology problems
Solutions to sample problems illustrating discrete dynamical systems in the context of biology. - Solutions to elementary discrete dynamical systems biology problems, part 2
Solutions to more sample problems illustrating discrete dynamical systems in the context of biology. - Elementary discrete dynamical systems biology problems
Sample problems illustrating discrete dynamical systems in the context of biology. - Elementary discrete dynamical systems biology problems, part 2
More sample problems illustrating discrete dynamical systems in the context of biology. - Solutions to elementary discrete dynamical systems problems
Solutions to sample problems illustrating discrete dynamical systems. - Solutions to elementary discrete dynamical systems problems, part 2
Solutions to more sample problems illustrating discrete dynamical systems. - Elementary discrete dynamical systems problems
Sample problems illustrating discrete dynamical systems.
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