 10.10.1.190: List the kinds of integrals in this chapter and how the integral th...
 10.10.1.191: How can work of a variable force be expressed by an integral?
 10.10.1.192: State from memory how you can evaluate a line integral. A double in...
 10.10.1.193: What do you remember about path independence? Why is it important?
 10.10.1.194: How did we Use Stokes's theorem in connection with path independence?
 10.10.1.195: State the definition of curl. Why is it important in this chapter?
 10.10.1.196: How can you transform a double integral or a surface integral into ...
 10.10.1.197: What is orientation of a surface? What is its role in connection wi...
 10.10.1.198: State the divergence theorem and its applications from memory.
 10.10.1.199: State Laplace's equation. Where in physics is it important? What pr...
 10.10.1.200: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.201: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.202: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.203: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.204: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.205: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.206: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.207: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.208: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.209: Evaluate. with F and C as given, by the method that seems most suit...
 10.10.1.210: Find the coordinmes .i. y of the center of gravity of a mass of den...
 10.10.1.211: Find the coordinmes .i. y of the center of gravity of a mass of den...
 10.10.1.212: Find the coordinmes .i. y of the center of gravity of a mass of den...
 10.10.1.213: Find the coordinmes .i. y of the center of gravity of a mass of den...
 10.10.1.214: Find the coordinmes .i. y of the center of gravity of a mass of den...
 10.10.1.215: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.216: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.217: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.218: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.219: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.220: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.221: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.222: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.223: Evaluate this integral directly or. if pos~ible. by the divergence ...
 10.10.1.224: Evaluate this integral directly or. if pos~ible. by the divergence ...
Solutions for Chapter 10: Vector Integral Calculus. Integral Theorems
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 10: Vector Integral Calculus. Integral Theorems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 10: Vector Integral Calculus. Integral Theorems includes 35 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. Since 35 problems in chapter 10: Vector Integral Calculus. Integral Theorems have been answered, more than 45587 students have viewed full stepbystep solutions from this chapter.

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

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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