 9.14.1: In 14, verify Stokes theorem. Assume that the surface S is oriented...
 9.14.2: In 14, verify Stokes theorem. Assume that the surface S is oriented...
 9.14.3: In 14, verify Stokes theorem. Assume that the surface S is oriented...
 9.14.4: In 14, verify Stokes theorem. Assume that the surface S is oriented...
 9.14.5: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.6: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.7: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.8: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.9: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.10: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.11: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.12: In 512, use Stokes theorem to evaluate C F d r. Assume C is oriente...
 9.14.13: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
 9.14.14: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
 9.14.15: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
 9.14.16: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
 9.14.17: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
 9.14.18: In 1316, use Stokes theorem to evaluate eeS (curl F) n dS. Assume t...
Solutions for Chapter 9.14: Stokes Theorem
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 9.14: Stokes Theorem
Get Full SolutionsChapter 9.14: Stokes Theorem includes 18 full stepbystep 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 stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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. AI 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.