 12.12.28: In each of 1 through 5, evaluate C F dR over any simple closed path...
 12.12.33: In each of 1 through 10, determine whether F is conservative in the...
 12.12.55: In each of 1 through 10, evaluate f (x, y,z)d.f (x, y,z) = x, is th...
 12.12.65: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.73: Let C be a simple closed path in the x, yplane with interior D. Le...
 12.12.76: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.86: In each of 1 through 5, use Stokess theorem to evaluate C F dR or (...
 12.12.93: Compute the scale factors for cylindrical coordinates. Use them to ...
 12.12.14: A particle moves once counterclockwise about the triangle with vert...
 12.12.29: In each of 1 through 5, evaluate C F dR over any simple closed path...
 12.12.34: In each of 1 through 10, determine whether F is conservative in the...
 12.12.56: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = y2 , i...
 12.12.66: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.74: Under the conditions of 1, show that D (2 2 ) d A = C y y dx + x x dy.
 12.12.77: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.87: In each of 1 through 5, use Stokess theorem to evaluate C F dR or (...
 12.12.94: Elliptic cylindrical coordinates are defined by x = a cosh(u) cos(v...
 12.12.15: A particle moves once counterclockwise around the circle of radius ...
 12.12.30: In each of 1 through 5, evaluate C F dR over any simple closed path...
 12.12.35: In each of 1 through 10, determine whether F is conservative in the...
 12.12.57: In each of 1 through 10, evaluate f (x, y,z)d. 1f(x, y,z) = 1, is t...
 12.12.67: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.75: Let C be a simple closed path in the x, yplane, with interior D. L...
 12.12.78: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.88: In each of 1 through 5, use Stokess theorem to evaluate C F dR or (...
 12.12.95: Elliptic cylindrical coordinates are defined by x = a cosh(u) cos(v...
 12.12.16: A particle moves once counterclockwise about the rectangle with ver...
 12.12.31: In each of 1 through 5, evaluate C F dR over any simple closed path...
 12.12.36: In each of 1 through 10, determine whether F is conservative in the...
 12.12.58: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = x + y,...
 12.12.68: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.79: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.89: In each of 1 through 5, use Stokess theorem to evaluate C F dR or (...
 12.12.96: Parabolic cylindrical coordinates are defined by x = uv, y = 1 2 (u...
 12.12.17: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.32: In each of 1 through 5, evaluate C F dR over any simple closed path...
 12.12.37: In each of 1 through 10, determine whether F is conservative in the...
 12.12.59: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z)= z, is t...
 12.12.69: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.80: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.90: In each of 1 through 5, use Stokess theorem to evaluate C F dR or (...
 12.12.5: In each of 1 through 10, evaluate the line integral.C F dR with F =...
 12.12.18: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.38: In each of 1 through 10, determine whether F is conservative in the...
 12.12.60: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = xyz, i...
 12.12.70: In each of 1 through 6, find the mass and center of mass of the she...
 12.12.81: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.91: Calculate the circulation of F=(x y)i+ x 2 yj+axzk counterclockwise...
 12.12.6: In each of 1 through 10, evaluate the line integral.C 4xy ds with C...
 12.12.19: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.39: In each of 1 through 10, determine whether F is conservative in the...
 12.12.61: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = y, is ...
 12.12.71: Find the flux of F = xi + yj zk across the part of the plane x + 2y...
 12.12.82: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.92: Calculate the circulation of F=(x y)i+ x 2 yj+axzk counterclockwise...
 12.12.7: In each of 1 through 10, evaluate the line integral.C F dR with F =...
 12.12.20: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.40: In each of 1 through 10, determine whether F is conservative in the...
 12.12.62: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = x 2 , ...
 12.12.72: Find the flux of F = x zi yk across the part of the sphere x 2 + y2...
 12.12.83: In each of 1 through 8, evaluate either F nd or M div(F) dV, whiche...
 12.12.8: In each of 1 through 10, evaluate the line integral.C yz ds with C ...
 12.12.21: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.22: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.41: In each of 1 through 10, determine whether F is conservative in the...
 12.12.63: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = z, is ...
 12.12.84: Let be a smooth closed surface and F a vector field that is continu...
 12.12.9: In each of 1 through 10, evaluate the line integral.C xyz dz with C...
 12.12.23: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.42: In each of 1 through 10, determine whether F is conservative in the...
 12.12.64: In each of 1 through 10, evaluate f (x, y,z)d. 1f (x, y,z) = xyz, i...
 12.12.85: Let be a piecewise smooth closed surface bounding a region M. Show ...
 12.12.10: In each of 1 through 10, evaluate the line integral.C xzdy with C g...
 12.12.24: In each of 4 through 11, use Greens theorem to evaluate C F dR. All...
 12.12.43: In each of 11 through 20, determine a potential function to evaluat...
 12.12.11: Find the work done by F = x 2 i 2yzj + zk in moving an object along...
 12.12.25: Let D be the interior of a positively oriented simple closed path C...
 12.12.44: In each of 11 through 20, determine a potential function to evaluat...
 12.12.12: Find the mass and center of mass of a thin, straight wire extending...
 12.12.26: Let u(x, y) be continuous with continuous first and second partial ...
 12.12.45: In each of 11 through 20, determine a potential function to evaluat...
 12.12.13: Show that any Riemann integral b a f (x)dx is a line integral C F d...
 12.12.27: Fill in the details of the following argument to prove Greens theor...
 12.12.46: In each of 11 through 20, determine a potential function to evaluat...
 12.12.47: In each of 11 through 20, determine a potential function to evaluat...
 12.12.48: In each of 11 through 20, determine a potential function to evaluat...
 12.12.49: In each of 11 through 20, determine a potential function to evaluat...
 12.12.50: In each of 11 through 20, determine a potential function to evaluat...
 12.12.51: In each of 11 through 20, determine a potential function to evaluat...
 12.12.52: In each of 11 through 20, determine a potential function to evaluat...
 12.12.53: Prove the law of conservation of energy, which states that the sum ...
 12.12.54: Complete the proof of Theorem 12.5 by filling in the details of the...
Solutions for Chapter 12: Vector Integral Calculus
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 12: Vector Integral Calculus
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This expansive textbook survival guide covers the following chapters and their solutions. Since 92 problems in chapter 12: Vector Integral Calculus have been answered, more than 7773 students have viewed full stepbystep solutions from this chapter. Chapter 12: Vector Integral Calculus includes 92 full stepbystep solutions.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.