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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.f = (x + 1)(2y - 1)
Find the components of the vector v with initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector u in the direction of v.P: (1, 1, 0), Q: (6, 2, 0)
What curves are represented by the following? Sketch them.[3 + 2 cos t, 2 sin t, 0]
What is a vector? A vector function? A vector field? A scalar? A scalar function? A scalar field? Give examples.
Find div v and its value at P.v = [x2, 4y2, 9z2], P: (-1, 0, 1/2)
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:a • b, b • a, b • c
List the definitions and most important facts and formulas for grad, div, curl, and ∇2. Use your list to write a corresponding report of 3–4 pages, with examples of your own. No proofs.
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = x2 - y2
A necessary and sufficient condition for positive definiteness of a quadratic form Q(x) = xTAx with symmetric matrix A is that all the principal minors are positive that is, Show that the form in
Transform to canonical form (to principal axes). Express [x1 x2]T in terms of the new variables [y1 y2]T.3.7x12 + 3.2x1x2 + 1.3x22 = 4.5
What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate
Show that the eigenvalues of a real matrix are real or complex conjugate in pairs.
Transform to canonical form (to principal axes). Express [x1 x2]T in terms of the new variables [y1 y2]T.4x12 + 24x1x2 - 14x22 = 20
Find further 2 × 2 and 3 × 3 matrices with multiple eigenvalues.
What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate
Find further 2 × 2 and 3 × 3 matrices with positive defect.
Find an eigenbasis and diagonalize.
Prove that A is normal if and only if the Hermitian and skew-Hermitian matrices in Prob. 18 commute.Data from Prob. 18This important concept denotes a matrix that commutes with its conjugate
Do there exist skew-symmetric orthogonal 3 × 3 matrices?
What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate
Find the matrix A in the linear transformation y = Ax, where x = [x1 x2]T (x = [x1 x2 x3]T) are Cartesian coordinates. Find the eigenvalues and eigenvectors and
Find the matrix A in the linear transformation y = Ax, where x = [x1 x2]T (x = [x1 x2 x3]T) are Cartesian coordinates. Find the eigenvalues and eigenvectors and
Powers of unitary matrices in applications may sometimes be very simple. Show that C12 = I in Example 2. Find further examples.
What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate
Find the matrix A in the linear transformation y = Ax, where x = [x1 x2]T (x = [x1 x2 x3]T) are Cartesian coordinates. Find the eigenvalues and eigenvectors and
Verify that A and Â = p-1AP have the same spectrum.
Let A = [αjk] be an n × n matrix with (not necessarily distinct) eigenvalues λ1, · · · , λn. Show.A - kI has the eigenvalues λ1 - k, · · ·, λn - k and the same eigenvectors as A.
Show that any square matrix may be written as the sum of a Hermitian and a skew-Hermitian matrix. Give examples.
Are the eigenvalues of A + B sums of the eigenvalues of A and of B?
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.
If not the whole output but only a portion of it is consumed by the industries themselves, then instead of Ax = x (as in Prob. 13), we have x - Ax = y, where x = [x1 x2 x3]T is produced, Ax
Find the eigenvalues. Find the eigenvectors.
Show that (ABC)̅ T = -C-1BA for any n × n Hermitain A, Skew-Hermitian B, and unitary C.
Showing all intermediate results, calculate the following expression or give reasons why they are undefined: Aa, AaT, (Ab)T, bTAT
Find and graph all values of:√81
Find all solutions and graph some of them in the complex plane.ez = 0
Are the following functions analytic? Use (1) or (7).f(z) = i/z8
What is d’Alembert’s solution method? To what PDE does it apply?
Solve by Laplace Transforms
Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph u(x, t) as in Fig. 291 in
Compare these PDEs with respect to general behavior of eigenfunctions and kind of boundary and initial conditions. State the difference between Fig. 291 in Sec. 12.3 and Fig. 295.
Why did we introduce polar coordinates in this section?
Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.f(x) = e-2x if x >
Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.f(x) = x if 1 < x
What is the Fourier transform? The discrete Fourier transform?
Summarize the second part of this section beginning with Complex Function, and indicate what is conceptually analogous to calculus and what is not.
Write a program for graphing equipotential lines u = const of a harmonic function u and of its conjugate v on the same axes. Apply the program to(a) u = x2 - y2, v = 2xy(b) u = x3 - 3xy2, v = 3x2y -
Find the principal value. Show details.(-3)3-i
Find and graph all values of:3√1
Find and graph all values of:4√-1
Find and graph all roots in the complex plane.3√216
Find the principal value. Show details.(1 + i)1-i
Determine α and b so that the given function is harmonic and find a harmonic conjugate.u = αx3 + bxy
Find the value of the derivative ofz3/(z + i)3 at i
Find and graph all roots in the complex plane.3√1 + i
Find the value of the derivative ofi(1 - z)n at 0
Solve for z.ln z = 0.6 + 0.4i
Determine α and b so that the given function is harmonic and find a harmonic conjugate.u = eπx cos αv
Let z = x + iy. Showing details, find, in terms of x and y:Im (1/z̅2)
Give the details of the derivative of (9).
Let z = x + iy. Showing details, find, in terms of x and y:Re (z/z̅), Im (z/z̅)
Find all solutions.sinh z = 0
Represent in polar form, with the principal argument.-15i
Write a program for calculating these roots and for graphing them as points on the unit circle. Apply the program to zn = 1 with n = 2, 3, · · ·, 10. Then extend the program to one for arbitrary
Find all solutions and graph some of them in the complex plane.ez = 1
Find the value of the derivative of(z - 4i)8 at = 3 + 4i
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).v = ex sin 2y
Let z = x + iy. Showing details, find, in terms of x and y:Re [(1 + i)16z2]
Let z = x + iy. Showing details, find, in terms of x and y:Re z4 - (Re z2)2
Find all solutions.cosh z = 0
Represent in polar form, with the principal argument.-4 - 4i
Graph in the complex plane and represent in the form x + iy:√8(cos 1/4π + i sin 1/4π)
Find out, and give reason, whether f (z) is continuous at z = 0 if f(0) = 0 and for z ≠ 0 the function f is equal to:(Re z)/(1 - |z|)
Show that the set of values of ln (i2) differs from the set of values of 2 ln i.
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).v = (2x + 1)y
Let z = x + iy. Showing details, find, in terms of x and y:Im (1/z), Im (1/z2)
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:4 (z1 + z2)/(z1 - z2)
Find, in the form x + iy, showing details,(1 + i)/(1 - i)
Graph in the complex plane and represent in the form x + iy:3 (cos 1/2π - i sin 1/2π)
Find out, and give reason, whether f (z) is continuous at z = 0 if f(0) = 0 and for z ≠ 0 the function f is equal to:|z2| Im (1/z)
Find Re and Im ofexp (z2)
Find all values and graph some of them in the complex plane.ln (ei)
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).u = x/(x2 + y2)
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:(z1 + z2)(z1 - z2), z12 - z22
Using the definitions, prove:cos z is even, cos (-z) = cos z, and sin z is odd, sin (-z) = -sin z.
Find, in the form x + iy, showing details,1/(4 + 3i)
Determine the principal value of the argument and graph it as in Fig. 325. (1 + i)20
Find and graph Re f, Im f, and |f| as surfaces over the z-plane. Also graph the two families of curves Re f(z) = const and Im f(z) = const in the same figure, and the curves |f(z)| = const in another
Find all values and graph some of them in the complex plane.ln 1
Write in exponential form (6):1 + i
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).u = xy
Determine the principal value of the argument and graph it as in Fig. 325. -π - πi
Write in exponential form (6):1/(1 - z)
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:(z1 - z2)2/16, (z1/4 - z2/4)2
Find, in the form u + iv,sin πi, cos (1/2π - πi)
Find, in the form x + iy, showing details,(2 + 3i)2
Determine the principal value of the argument and graph it as in Fig. 325. 3 ± 4i
Find Re f, and Im f and their values at the given point z.f (z) = 1/(1 - z) at 1 - i
Find Ln z when z equalsei
Write in exponential form (6):-6.3

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