 10.4.10.1.61: Using Green's theorem, evaluate f F(r)drcounterclockwise c around ...
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 10.4.10.1.73: Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In bound...
 10.4.10.1.74: Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In bound...
 10.4.10.1.75: Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In bound...
 10.4.10.1.76: Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In bound...
 10.4.10.1.77: CAS EXPERIMENT. Apply (4) to figures of your choice whose area can ...
 10.4.10.1.78: (Laplace's equation) Show that for a solution w(x, y) of Laplace's ...
 10.4.10.1.79: Show that w = 2ex cos)' satisfies Laplace's equation V2 w = 0 and. ...
 10.4.10.1.80: PROJECT. Other Forms of Green's Theorem in the Plane. Let Rand C be...
Solutions for Chapter 10.4: Green's Theorem in the Plane
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 10.4: Green's Theorem in the Plane
Get Full SolutionsSince 20 problems in chapter 10.4: Green's Theorem in the Plane have been answered, more than 46265 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. 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. Chapter 10.4: Green's Theorem in the Plane includes 20 full stepbystep solutions.

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

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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