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Textbooks / Math / Differential Equations and Boundary Value Problems: Computing and Modeling 5

Differential Equations and Boundary Value Problems: Computing and Modeling 5th Edition - Solutions by Chapter

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition

ISBN: 9780321796981

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 416 Reviews
Textbook: Differential Equations and Boundary Value Problems: Computing and Modeling
Edition: 5
Author: C. Henry Edwards, David E. Penney, David T. Calvis
ISBN: 9780321796981

Since problems from 58 chapters in Differential Equations and Boundary Value Problems: Computing and Modeling have been answered, more than 73286 students have viewed full step-by-step answer. The full step-by-step solution to problem in Differential Equations and Boundary Value Problems: Computing and Modeling were answered by , our top Math solution expert on 01/04/18, 09:22PM. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This expansive textbook survival guide covers the following chapters: 58. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5.

Key Math Terms and definitions covered in this textbook
  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Outer product uv T

    = column times row = rank one matrix.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.