×
Log in to StudySoup
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
×
Reset your password

# Solutions for Chapter 14.1: Line Integral in the Complex Plane

## Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Solutions for Chapter 14.1: Line Integral in the Complex Plane

Solutions for Chapter 14.1
4 5 0 286 Reviews
19
2
##### ISBN: 9780471488859

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Since 35 problems in chapter 14.1: Line Integral in the Complex Plane have been answered, more than 49588 students have viewed full step-by-step solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter 14.1: Line Integral in the Complex Plane includes 35 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Condition number

cond(A) = c(A) = IIAIlIIA-III = 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.

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Distributive Law

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

• Hypercube matrix pl.

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

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Left inverse A+.

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

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• 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.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > 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).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Vector addition.

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

×
Log in to StudySoup
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
×
Reset your password