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# Solutions for Chapter 4.8: Multiple Integrals

## Full solutions for Numerical Analysis | 10th Edition

ISBN: 9781305253667

Solutions for Chapter 4.8: Multiple Integrals

Solutions for Chapter 4.8
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##### ISBN: 9781305253667

Chapter 4.8: Multiple Integrals includes 20 full step-by-step solutions. Numerical Analysis was written by Patricia and is associated to the ISBN: 9781305253667. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Since 20 problems in chapter 4.8: Multiple Integrals have been answered, more than 3582 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

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

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Normal matrix.

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

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Sum V + W of subs paces.

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

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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