 18.18.41: Solve 2 u(x, y,z) = 0 for 0 < x < 1, 0 < y < 1, 0 < z < 1, u(0, y,z...
 18.18.45: In each of 1 through 4, write a solution for the steadystate tempe...
 18.18.1: Show that if f and g are harmonic on D so are f + g and, for any co...
 18.18.3: In each of 1 through 5, solve the Dirichlet problem for the given r...
 18.18.12: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.24: In each of 1 through 4, find an integral formula for the solution o...
 18.18.29: Write an integral solution for the Dirichlet problem for the upper ...
 18.18.30: Write an integral solution for the Dirichlet problem for the upper ...
 18.18.42: Solve 2 u(x, y,z) = 0 for 0 < x < 2, 0 < y < 2, 0 < z < 1, u(x, y, ...
 18.18.46: In each of 1 through 4, write a solution for the steadystate tempe...
 18.18.2: Show that the following functions are harmonic (on the entire plane...
 18.18.4: In each of 1 through 5, solve the Dirichlet problem for the given r...
 18.18.13: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.25: In each of 1 through 4, find an integral formula for the solution o...
 18.18.31: Write an integral solution for the Dirichlet problem for the right ...
 18.18.43: Solve 2 u(x, y,z) = 0 for 0 < x < 1, 0 < y < 2, 0 < z < , u(0, y,z)...
 18.18.47: In each of 1 through 4, write a solution for the steadystate tempe...
 18.18.5: In each of 1 through 5, solve the Dirichlet problem for the given r...
 18.18.14: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.26: In each of 1 through 4, find an integral formula for the solution o...
 18.18.32: Write an integral solution for the Dirichlet problem for the right ...
 18.18.44: Solve 2 u(x, y,z) = 0 for 0 < x < 1, 0 < y < 2, 0 < z < , u(x, 0,z)...
 18.18.48: In each of 1 through 4, write a solution for the steadystate tempe...
 18.18.6: In each of 1 through 5, solve the Dirichlet problem for the given r...
 18.18.15: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.27: In each of 1 through 4, find an integral formula for the solution o...
 18.18.33: Find a general formula for the solution of the Dirichlet problem fo...
 18.18.49: Solve for the steadystate temperature distribution in a hollowedo...
 18.18.7: In each of 1 through 5, solve the Dirichlet problem for the given r...
 18.18.16: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.28: Show that, for 0 r < 1, r n sin(n )= 1 2 1 r 2 1 +r 2 2r cos( ) sin...
 18.18.34: Write an integral solution for the Dirichlet problem for the lower ...
 18.18.50: 2 u(x, y)= 0 for 0 < x < 1, 0 < y < 1, u y (x, 0)= 4 cos(x), u y (x...
 18.18.8: Apply separation of variables to the problem 2 u = 0 for 0 < x < a,...
 18.18.17: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.35: Find the steadystate temperature distribution in a thin, homogeneo...
 18.18.51: 2 u(x, y)= 0 for 0 < x < 1, 0 < y < , u y (x, 0)= u y (x,) = 0 for ...
 18.18.9: Use separation of variables to solve 2 u = 0 for 0 < x < a, 0 < y <...
 18.18.18: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.36: Solve the Dirichlet problem for the strip < x < , 0< y <1 if u(x, 0...
 18.18.52: 2 u(x, y)= 0 for 0 < x < , 0 < y < , u y (x, 0)= cos(3x) for 0 x u ...
 18.18.10: Solve for the steadystate temperature distribution in a homogeneou...
 18.18.19: In each of 1 through 8, write the solution of the Dirichlet problem...
 18.18.37: Solve for the steadystate temperature distribution in an infinite,...
 18.18.53: Use separation of variables to write an expression for the solution...
 18.18.11: Solve for the steadystate temperature distribution in a thin, flat...
 18.18.20: In each of 9 through 12, solve the problem by converting it to pola...
 18.18.38: Solve the following problem: 2 u(x, y) = 0 for 0 < x < , 0 < y < 2 ...
 18.18.54: Attempt a separation of variables to solve 2 u(x, y) = 0 for 0 < x ...
 18.18.21: In each of 9 through 12, solve the problem by converting it to pola...
 18.18.39: Solve for the steadystate temperature distribution in a homogeneou...
 18.18.55: Write a series solution for 2 u(r,) = 0 for 0 r < R, u r (R,) = sin...
 18.18.22: In each of 9 through 12, solve the problem by converting it to pola...
 18.18.40: Write a general expression for the steadystate temperature distrib...
 18.18.56: Write a series solution for 2 u(r,) = 0 for 0 r < R, u r (R,) = cos...
 18.18.23: In each of 9 through 12, solve the problem by converting it to pola...
 18.18.57: Solve the following Neumann problem for the upper halfplane: 2 u(x...
 18.18.58: Solve the following Neumann problem for the upper halfplane: 2 u(x...
 18.18.59: Solve the following Neumann problem for the upper halfplane: 2 u(x...
 18.18.60: Solve the following Neumann problem for the right quarterplane: 2 ...
 18.18.61: Solve the following mixedboundary value problem: 2 u(x, y) = 0 for...
Solutions for Chapter 18: The Potential Equation
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 18: The Potential Equation
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This expansive textbook survival guide covers the following chapters and their solutions. Since 61 problems in chapter 18: The Potential Equation have been answered, more than 7731 students have viewed full stepbystep solutions from this chapter. Chapter 18: The Potential Equation includes 61 full stepbystep solutions.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

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

lAII = 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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = 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 ().

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.