Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 14 - Index

# Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 14 - Index

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Guide to Commonly Used Notation Symbol Usual Meaning R the set of real numbers R n n-dimensional Euclidean space t independent variable (time) u , v vectors consisting of real numbers x , y vectors consisting of dependent variables x , y vectors consisting of initial conditions x ( ) , y ( ) x * , y * equilibrium solution of a system of ode s f a vector consisting of functions A , M , P square n × n matrices D a diagonal n × n matrix N a nilpotent n × n matrix tr ( A ) trace of a square matrix A det ( A ) determinant of a square matrix A λ eigenvalue of a matrix, or a Lyapunov exponent α , β real, imaginary parts of an eigenvalue λ = α + β i E s , E u , E c stable, unstable, and center subspaces W s , W u , W c stable, unstable, and center manifolds A ⊕ B direct sum of subspaces A and B span { v 1 , v 2 , . . . v n } span of vectors v 1 , v 2 , . . . v n φ t ( x ) solution of x = f ( x ) with x ( ) = x φ t flow of a system of ode s f : R n → R m a function from R n into R m ∇ f gradient of a function from R n into R J f Jacobian matrix of a function from R n into R m u • v dot product of vectors u and v v 2 Euclidean norm (length) of a vector v in R n u- v 2 Euclidean distance between points u and v in R n B ( x , ) open ball of radius centered at x V ( x ) a Lyapunov function Γ ( t ) a parametrized curve, Γ : R → R n 404 guide to commonly used notation 405 Symbol Usual Meaning n a normal vector to a parametrized curve r , θ radius and angle (polar coordinates) μ a bifurcation parameter σ Lyapunov number (associated with Hopf bifurcations) τ time delay (for delay differential equations) x ( a + ) the right-hand limit lim t → a + x ( t ) x ( a- ) the left-hand limit lim t → a- x ( t ) x * a fixed-point of a first-order difference equation Sf ( x ) Schwarzian derivative of a function f : R → R γ feedback gain parameter in the tdas algorithm L linear operator, or length of the interval [ 0, L ] φ ( x ) initial condition for a pde ∂ Ω boundary of the domain Ω κ positive-valued diffusion constant (heat equation) c wave speed (wave/transport equations) ψ ( x ) initial velocity (wave equation) S φ convolution of functions S and φ S ( x , t ) (one-dimensional) heat kernel φ odd , φ even odd, even extensions of a function φ X ( x ) , Y ( y ) , T ( t ) separated solutions of pde s (separation of variables) R ( r ) , Θ ( θ ) separated solutions of pde s (polar coordinates) β abbreviation for square roots of eigenvalues λ A n , B n Fourier coefficients f ∞ , f L 2 different types of norms of the function f f , g inner product of functions f and g S N ( x ) partial sum of an infinite series of functions f ( x- ) , f ( x + ) left and right-hand limits of f ( x ) as x → x f ( x- ) , f ( x + ) left and right-hand derivatives of f ( x ) as x → x Δ Laplace operator v ( r ) a radial solution of Laplace’s or Poisson’s equation h ( θ ) boundary condition for Laplace’s equation on a disc References [ 1 ] G. Bachman, L. Narici, and E. Beckenstein,] G....
View Full Document

## This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

### Page1 / 7

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 14 - Index

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online