Problem Sheet1

Course Title - Engineering Mathematics and Computation (3-0-2)

Course number - APL703, Credits - 4

Semester-I (2015)

by Arghya Samanta

Department of Applied Mechanics, IIT Delhi

Vector Calculus (2 Lectures)

Contents:

Basic idea of vectors and its algebric properties, Unit vector, Scalar or dot product, Cauchy In-equality, Vector cross product, Differentiation of vectors, Gradient, Divergence, Curl, Solenoidalvector, Irrotational vector, Gauss Divergence theore, Stokes theorem, Curvature.

Problems sheet:

Problem 1: Suppose ~a, ~b and ~c are three vectors, then prove that ~a (~b ~c) = (~a~b) ~c.

Problem 2: Find a unit vector normal to the surface (x2 + y2 z) = 1 at the point P (1, 1, 1).

Problem 3: If is a scalar and ~v is a vector, then prove that ~ is an irrotational vector and ~~vis a solenoidal vector.

Problem 4: Suppose ~v is a vector, then prove that ~ (~ ~v) = ~(~ ~v) ~2~v.

Problem 5: Consider an incompressible homogeneous viscous fuid with density . Then prove that~ ~v = 0, where ~v is a velocity vector of the liquid.

Problem 6: A vector ~f whose flux over every closed surface vanishes, then prove that ~f is solenoidal.

Problem 7: Let ~f = (ye1 + xe2) and d~r = dxe1 + dye2. Applying Stokes theorem find the areaof a ellipse.

Problem 8: Show that the curvature of a curve y = f(x) at any point x is = f /(1 + f 2)3/2.

1

Tensor Calculus (2 Lectures)

Contents:

Index notation, Einstiens summation convention, Kronecker Delta, Permutation symbol, Vectorsrepresentation in terms of index notation, Contraction, Definition of tensor and its componentsin matrix form, Identity tensor, Zero tensor, Prouct of two tensors, Dyadic product, Transposeof tensor, Trace of tensor, Orthogonal tensor, Transformation of tensor between two rectrangularcartesian coordinate systems, Order of tensor, Symmetric and Antisymmetric tensors, Dual vector,Coordinate transformation.

Problems sheet:

Problem 1: Show that the Kronecker Delta matrix [ij ] is nothing but a identity matrix. For aNewtonian fluid, the stress tensor is defined as ij = pij + 2Eij . Find ii?

Problem 2: Given that a tensor T transforms the base vectors {ei} as follows

T e1 = 2e1 6e2 + 4e3

T e2 = 3e1 + 4e2 e3

T e3 = 2e1 + e2 + 2e3

How does this tensor T transform the vector ~a = e1 + 2e2 + 3e3?

Problem 3: Suppose the tensor R corresponds to a right hand rotation of a rigid body about x3axis by an angle . Find the tensor R and check its orthogonality.

Problem 4: Given that the component of a vector ~a with respect to the base vector {ei} are givenby (2, 0, 0). Find its component with respect to the base vectors {ei}, where e

i axes are obtained

by a 90 counter-clockwise rotation of the ei axes about e3 axis.

Problem 5: If in 2D cartesian coordinate system ~ = (x, y). Find ~~v and ~ ~v in polar coor-dinate system (r, ).

problem 6: If in 3D cartesian coordinate system ~ = (x, y, z). Find ~~v and ~~v in cylindricalcoordinate system (r, , z).

problem 7: If in 3D cartesian coordinate system ~ = (x, y, z). Find ~~v and ~ ~v in sphericalpolar coordinate system (r, , ).

2

Linear Algebra (5 Lectures)

Contents:

Linear system of equations, homogeneous and nonhomogeneous systems, Cramers rule, GaussElimination, Row equivalence, Row reduced echelon matrix, Row rank, Column rank, Vectorspace, Subspace, Linear combination, Linear independent, Basis, Dimension, Inner product, Norm,Pythagorous law, Schwarz Inequality, Triangle law, Orthogonal decomposition, Linear transforma-tion, Null space, Image space, Characteristic polynomial, Eigenvalue, Eigenvector, Eigen space,Algebric multiplicity, Geometric multiplicity, Spectrum, Diagonalization, Quadratic form, Canon-ical form, Positive definite, Negative definite.

Problems sheet:

Problem 1: Solve the following system using Crammers rule

x1 2x2 + x3 = 1

2x1 + x2 2x3 = 3

x1 + 3x2 + 4x3 = 2

Problem 2: Solve the following system using Gauss elimination

x2 + 2x3 = 3

2x1 + 4x2 + 8x3 + 2x4 = 4

x1 + 2x2 + 4x3 + 2x4 = 2

x1 + x2 + 6x3 + x4 = 5

Problem 3: Prove that a vector space V has a unique identity element and unique inverse element.

Problem 4: Suppose U and W are two subspaces of V . Show that U +W is a subspace of V .Further show that U +W is a smallest subspace containing both U and W .

Problem 5: Show that the following vectors {(0, 3, 4), (1, 1, 1), (1, 0, 1)} are basis of R3.

Problem 6: Let S be the set of vectors in P2 (set of real polynomials of degree 2) which are of theform at2 + b. Then Show that S is a subspace of P2. Find also the dimension of S.

Problem 7: Consider the subspace S of M22 (set of all 2 2 matrices) which consists of matrices

of the form

(a bc d

). Find the basis of S and evaluate the dimension of S.

Problem 8: If any two vectors u and v in a vector space V are orthogonal to each other, then provethat u+ v2 = u2 + v2.Problem 9: For any two vectors u and v in a vector space V , show that |u, v| uv.

Problem 10: Prove that set of orthogonal nonzero vectors are linearly independent.

Problem 11: A linear transformation T : R3 R2 is given by T (x) = Ax, where

(1 2 34 5 6

).

Find KerT or null space of T and evaluate the nullity of T .

Problem 12: Show that if the determinant of a matrix is zero, then one of the eigenvalue of thematrix is zero.

Problem 13: If is an eigenvalue of a nonsingular matrix. Then 1 is an eigenvalue of A1.

3

Problem 14: Determine the eigenvalues and eigenvectors of the matrix A =

2 1 13 2 3

3 1 2

. Fur-

ther check that 1 + 2 + 3 = tr(A).

Problem 15: Show that the eigenvalues of the real orthogonal matrix are of unit modulus.

Problem 16: Diagonalize the matrix A =

2 1 13 2 3

3 1 2

. Find the matrix A3.

Problem 17: Reduce the quadratic form of the following expression to its standard form involvingthe principal axes O{y1, y2}, where P (x1, x2) = x

21 + 4x1x2 + 4x

22.

4

Solution of nonlinear equations (1 Lecture)

Contents:

Newton Raphson method, Its advantage and disadvantage, Its convergence.

Problems sheet:

Problem 1: Use Newtons method to find the solution of y = f(x) = x 2 sinx up to O(107).

Problem 2: Let f(x) = x2 a. Show that the Newton Method leads to the recurrence relation

xn+1 =12

(xn +

axn

)

5

Variational Calculus (5 Lectures)

Contents:

Introduction of Functional, Shortest Distance, Brachistochrone Problem, Minimal Surface Area,Function Space, Repitition of Normed Linear Space, Linear Functional, Variation of functional,Euler-Lagrange Equation, Problem with Constraints, Multivariate Case, Lagrange Multiplier

Problems sheet:

Problem 1: Find the shortest plane curve joining two points A and B, i.e. find the curve y = y(x)

for which the functional ba

1 + y2dx achieves its minimum.

Problem 2: Let A and B be two fixed points. Then the time it takes a particle to slide under theinfluence of gravity along some path joining A and B depends on the choice of path. Determinethe path for which particle takes least time to go from A to B.

Problem 3: Among all curves joining two points (x0, y0) and (x1, y1), find the one which generatesthe surface of minimum area when rotated about the axis.

Problem 4: Suppose that J [y] = 21

1 + y2/xdx, with y(1) = 0, y(2) = 1. Find the solution y(x)

for which J has an extremum.

Problem 5: Test for an extremum of the functional J [y] = 10(xy + y2 2y2y)dx, with y(0) =

1, y(1) = 2.

Problem 6: Which curve minimizes the functional J [y] = 10(y2/2 + yy + y + y)dx, when the

values of y are not specified at the end point?

Problem 7: Determine the closed curve C (has a fixed length) which encloses a maximum area.

6

Fourier Series and Fourier Transform (5 Lecture)

Contents:

Introduction, Even function, Odd function, Fourier Coefficients, Periodic function, Representationof a function in Fourier series, Representation of an even function in Fourier series, Representationof an odd function in Fourier series, Fourier Transform, Inverse Fourier Transform, Linear Propertyof Fourier Transform, Fourier Transform of Derivatives, Fourier Transform of Partial Derivatives,Convolution of Two Functions.

Problems sheet:

Problem 1: Find the Fourier series corresponding to the function defined as

f(x) =

sin 2x, if, x < /2

0, else if, /2 x 0

sin 2x, else, 0 < x

Problem 2: If f(x) is an even function, show that the Fourier series representation of f is given by

f(x) = a0 +

n=1

an cosnx,

where a0 =1pi

pi0f(x)dx, and an =

2pi

pi0f(x) cosnxdx.

Problem 3: If f(x) is an odd function, show that the Fourier series representation of f is given by

f(x) =

n=1

bn sinnx,

where bn =2pi

pi0f(x) sinnxdx.

Problem 4: Find the Fourier series corresponding to the function f(x) = |x|, in the intervalL x L.Problem 5: Find the Fourier transform of exp(ax2).Problem 6: Show that the Fourier transform of a Gaussian distribution is itself Gaussian.Problem 7: Find the temperature distribution corresponding to the Heat equation

1

T

t=

2T

x2,

where t > 0, T (x, t) 0 as |x| and T (x, 0) = f(x), < x a.

Problem 9: Find the Fourier transform of exp(a|x|), where a > 0.

Problem 10: Using Fourier transform, solve the following differential equation

du

dx+ 2u = exH(x),

where H(x) is a Heaviside step function, defined as

H(x) =

{1, if, x 0

0, else if, x 0.

7

Problem 11: Find the solution of the wave equation

2u(x, t)

x2=

1

c22u(x, t)

t2, t > 0, < x

Differential Equations (8 Lectures)

Contents:

Analytical solution of 1st and 2nd order ODEs, Integrating Factor, Homogeneous and nonHomoge-neous ODEs, Linear ODE, Bernouliis Equation, Method of undetermined coefficients, Variation ofParameter, Euler method, Runge-Kutta Method, Finite difference method, Sturm-Liouville prob-lem.

Problems sheet:

Problem 1: Find the solution of the first order differential equation

(3x+ 2y)dx+ (2x+ y)dy = 0.

Problem 2: Show that eP (x)dx is an integrating factor the linear first order differential equation

dy

dx+ P (x)y = Q(x).

Problem 3: Transform the Bernoullis equation

dy

dx+ P (x)y = Q(x)yn

into a suitable linear equationdz

dx+R(x)z = S(x)

and write R(x) and S(x) in terms of P (x) and Q(x).

Problem 4: Solve the following first order linear ODE

(x2 + 1)dy

dx+ 4xy = x.

Problem 5: Solve the following first order linear ODE

dy

dx+ y = xy3.

Problem 6: Solve the following second order linear ODE

d2y

dx2 6

dy

dx+ 25y = 0, y(0) = 3, y(0) = 1.


d2y

dx2 3

dy

dx+ 2y = x2ex.


d2y

dx2 2

dy

dx 3y = 2ex 10 sinx.


d2y

dx2+ y = tanx.

9

Problem 10: Solve the following ODE

x2d2y

dx2 2x

dy

dx+ 2y = x3.

Problem 11: Transform the following ODE into linear equation with constant coefficients

a0x2 d

2y

dx2+ a1x

dy

dx+ a2y = F (x),

where a0, a1 and a2 are constants.Problem 12: Solve the following ODE using finite difference method

d2x

dt2+ x(t) = 0, x(0) = 1,

dx

dt(0) = 0,

where space step t = 0.05. Find the solution at t = 0.1 and compare with the analytical solution.How much error do you expect?

Problem 13: Solve the following ODE using Euler, improved Euler and Runge-Kutta methods andcompare solutions with the exact one at t = 0.1, where step length t = 0.1.

dy

dt+ 2y = 2 e4t, y(0) = 1.

Problem 14: Solve the following eigenvalue problem and determine the corresponding eigenvaluesand eigenfunctions

d2y

dx2+ y = 0, y(0) = 0, y() = 0.

Problem 15: Suppose (x) and (x) are two distinct eigen functions corresponding to the eigen-values and of the following Strum-Liouville problem

d

dx

[p(x)

dy

dx

]+ [q(x) + r(x)]y = 0, y(a) = 0, y(b) = 0.

Show that and are orthogonal.

10

References:

1. G. Strang, Differential Equations and Linear Algebra

2. G. Strang, Introduction to Applied Mathematics

3. G. Strang, Computational Science and Engineering

4. I. M. Gelfand and V. Fomin, Calculus of Variations

5. M. T. Heath, Scientific Computing

6. A. Jeffrey, Advanced Engineering Mathematics

7. K. Singh, Linear Algebra

8. S. Axler, Linear Algebra

9. L. Debnath, Integral Transforms and Their Applications.

10. S. L. Ross, Differential Equations.

11. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes inFortran 77.

Grade Evaluation of the Course:

(i) 15% weightage will be given on the basis of performance in each of the two minor tests (1-hourduration) to be held during the semester according to the time-table provided by the AcademicSection.

(ii) 35% weightage will be given on the basis of performance in the 2-hour major test to be heldat the end of the semester.

(iii) 15% weightage will be given on the basis of performance in the Practice Session. (a) Twohome assignment and (b) Performance in theoretical Quizze session

(iv) 20% weightage will be given on the basis of performance in the computational work.

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Problem Sheet1