 10.9.10.1.170: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.171: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.172: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.173: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.174: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.175: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.176: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.177: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.178: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.179: Evaluate the integral II (curl F) 0 n dA directly for the given F a...
 10.9.10.1.180: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.181: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.182: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.183: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.184: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.185: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.186: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.187: Calculate this line integral by Stokes's theorem, clockwise as seen...
 10.9.10.1.188: (Stokes's theorem not applicable) Evaluate f Fo r' ds, c F = (x2 + ...
 10.9.10.1.189: WRITING PROJECT. Grad, Div, Curl in Connection with Integrals. Make...
Solutions for Chapter 10.9: Stokes's Theorem
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 10.9: Stokes's Theorem
Get Full SolutionsSince 20 problems in chapter 10.9: Stokes's Theorem have been answered, more than 46098 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 10.9: Stokes's Theorem includes 20 full stepbystep solutions.

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.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.