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# Solutions for Chapter 9.14: Stokes Theorem

## Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Solutions for Chapter 9.14: Stokes Theorem

Solutions for Chapter 9.14
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##### ISBN: 9781284105902

Chapter 9.14: Stokes Theorem includes 18 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since 18 problems in chapter 9.14: Stokes Theorem have been answered, more than 39229 students have viewed full step-by-step solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

Key Math Terms and definitions covered in this textbook
• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

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

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

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• 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.

• Length II x II.

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

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• 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.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• 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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