 9.9.9.1.269: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.270: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.271: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.272: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.273: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.274: Find curl v for v given with respect to righthanded Cartesian coor...
 9.9.9.1.275: What direction does curl v have if v is a vector parallel to the xz...
 9.9.9.1.276: Prove Theorem 2. Give two examples for (2) and (3) each.
 9.9.9.1.277: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.278: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.279: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.280: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.281: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.282: Let v be the velocity vector of a steady fluid flow. Is the flow ir...
 9.9.9.1.283: WRITING PROJECT. Summary on Grad, Div, Curl. List the definition an...
 9.9.9.1.284: PROJECT. Useful Formulas for the Curl. Assuming sufficient differen...
 9.9.9.1.285: With respect to righthanded coordinates, let u = [y2, .;:2, x 2], ...
 9.9.9.1.286: With respect to righthanded coordinates, let u = [y2, .;:2, x 2], ...
 9.9.9.1.287: With respect to righthanded coordinates, let u = [y2, .;:2, x 2], ...
 9.9.9.1.288: With respect to righthanded coordinates, let u = [y2, .;:2, x 2], ...
Solutions for Chapter 9.9: Curl of a Vector Field
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 9.9: Curl of a Vector Field
Get Full SolutionsSince 20 problems in chapter 9.9: Curl of a Vector Field have been answered, more than 44379 students have viewed full stepbystep solutions from this chapter. Chapter 9.9: Curl of a Vector Field includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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 equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.