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 9.21: Find the velocity and acceleration of a particle whose position vec...
 9.22: The velocity of a moving particle is v(t) 10t i (3t 4t) j k. If the...
 9.23: The acceleration of a moving particle is a(t) "2 sin t i "2 cos t j...
 9.24: Given that r(t) 1 2t 2 i 1 3t 3 j 1 2t 2 k is the position vector o...
 9.25: Sketch the curve traced by r(t) cosh t i sinh t j t k.
 9.26: Suppose that the vector function of is the position vector of a mov...
 9.27: In 27 and 28, find the directional derivative of the given function...
 9.28: In 27 and 28, find the directional derivative of the given function...
 9.29: Consider the function f (x, y) x2 y4 . At (1, 1) what is (a) The ra...
 9.30: Let w "x2 y2 z 2 . (a) If x 3 sin 2t, y 4 cos 2t, and z 5t 3 , find...
 9.31: Find an equation of the tangent plane to the graph of z sin (xy) at...
 9.32: Determine whether there are any points on the surface z 2 xy 2x y2 ...
 9.33: Express the volume of the solid shown in FIGURE 9.R.1 as one or mor...
 9.34: A lamina has the shape of the region in the first quadrant bounded ...
 9.35: Find the moment of inertia of the lamina described in about the ya...
 9.36: Find the volume of the sphere x2 y2 z 2 a2 using a triple integral ...
 9.37: Find the volume of the solid that is bounded between the cones z "x...
 9.38: Find the volume of the solid shown in FIGURE 9.R.2.
 9.39: In 3942, find the indicated expression for the vector field F x2 y ...
 9.40: In 3942, find the indicated expression for the vector field F x2 y ...
 9.41: In 3942, find the indicated expression for the vector field F x2 y ...
 9.42: In 3942, find the indicated expression for the vector field F x2 y ...
 9.43: Evaluate # C z 2 x2 y2 ds, where C is given by x cos 2t, y sin 2t, ...
 9.44: Evaluate C (xy 4x) ds, where C is given by 2x y 2 from (1, 0) to (0...
 9.45: Evaluate C 3x2 y2 dx (2x3 y 3y2 ) dy, where C is given by y 5x4 7x2...
 9.46: Show that BC y dx x dy x2 y2 2p , where C is the circle x2 y2 a2 .
 9.47: Evaluate C y sin pz dx x2 ey dy 3xyz dz, where C is given by x t, y...
 9.48: If F 4y i 6x j and C is given by x2 y2 1, evaluate C F dr in two di...
 9.49: Find the work done by the force F x sin y i y sin x j acting along ...
 9.50: Find the work done by F 2 x2 y2 i 1 x2 y2 j from (1 2, 1 2) to (1, ...
 9.51: Evaluate S (z/xy) dS, where S is that portion of the cylinder z x2 ...
 9.52: If F i 2 j 3 k, find the flux of F through the square defined by 0 ...
 9.53: If Fc(1/r), where c is constant and r r, rx iy jz k, find the f...
 9.54: Explain why the divergence theorem is not applicable in 53.
 9.55: Find the flux of F c(1/r) through any surface S that forms the boun...
 9.56: If F 6x i 7z j 8y k, use Stokes theorem to evaluate S (curl F n) dS...
 9.57: Use Stokes theorem to evaluate C 2y dx 3x dy 10z dz, where C is the...
 9.58: Find the work C F dr done by the force F x2 i y2 j z 2 k around the...
 9.59: If F x i y j z k, use the divergence theorem to evaluate S (F n) dS...
 9.60: Repeat for F 1 3x3 i 1 3y3 j 1 3z 3 k.
 9.61: If F (x2 ey tan1 z) i (x y) 2 j (2yz x10) k, use the divergence the...
 9.62: Suppose F x i y j (z 2 1) k and S is the surface of the region boun...
 9.63: Evaluate the integral R (x2 y2 ) "3 x2 2 y2 dA, where R is the regi...
 9.64: Evaluate the integral 6 R 1 "(x 2 y) 2 2(x y) 1 dA, where R is the ...
 9.65: As shown in FIGURE 9.R.4, a sphere of radius 1 has its center on th...
 9.66: On the surface of a globe or, more precisely, on the surface of the...
Solutions for Chapter 9: Vector Calculus
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 9: Vector Calculus
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9: Vector Calculus includes 66 full stepbystep solutions. Since 66 problems in chapter 9: Vector Calculus have been answered, more than 36397 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.