Courses Catalogue

Engineering Mathematics Ii

ACADEMIC PROGRAMME: Civil Engineering, Bsc
COLLEGE/SCHOOL/FACULTY: School of Engineering and Applied Sciences
PROGRAMME TYPE: Undergraduate

Course Content and Outline



Unit I: Linear Algebra

Eigenvalue problem                                                                                                (6 Hours )

Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues and Eigenvectors – Eigenbases– Cayley-Hamilton theorem (excluding proof) - Similarity transformation (Concept only) – Orthogonal transformation of a symmetric matrix to diagonal form – Quadratic form – Orthogonal reduction to its canonical form.


Vector spaces                                                                                                           (8 Hours)

Definition, Examples – Subspaces – Linear Span – Inner product & spaces– Linear Independence –Linear Dependence –Norms- Basis – Dimension– Orthogonal and Orthonormal Sets – Orthogonal Basis– Orthonormal Basis – Gram-Schmidt orthogonalisation process – Inner product spaces –Definition – Examples – Inequalities ; Schwartz, Triangle (No proof). Decomposition of a vector with respect to a subspace and its orthogonal complement – Pythagoras Theorem.


Unit II: Applications of Complex Numbers                                                       (6 Hours)

Applications of De Moivre's theorem, Exponential, Circular, Hyperbolic and Logarithmic functions of a complex variable, Inverse Hyperbolic functions, Real and imaginary parts of Circular and Hyperbolic functions, Summation of the seriesC+iS' method.


Unit III: Gradient, Divergence and Curl                                                            (6 Hours)

Gradient, Divergence and Curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green's theorem in a plane, Gauss divergence theorem and Stoke's theorem (excluding proof) – Simple applications involving cubes and rectangular parallelepipeds.


Unit IV: Differential Equations (DE) and Applications.

Differential Equation

Introduction                                                                                                           (4Hours)

Definition of a Differential Equation, Formation of a Differential Equation, Ordinary and Partial Differential Equations, Order and Degree of a Differential Equation.

Equation of first Order and first Degree                                                          (8 Hours)

Solution of different types of equations: (i) Variable separable (ii) Homogeneous Equations (iii) Equation reducible to homogeneous form (iv)Linear equations (v) Exact Differential Equations. Concept of Particular and General Solution, Solving by Substitution

Linear Differential Equations:                                                                             (6 Hours)

With constant coefficients of orders two: Definition, complete solution Rules for finding the complementary function. Rules for finding the particular Integral, Simple Problems. Lagrange’s linear partial differential equation

ODEs:                                                                                                                        (6 Hours)

Solution of higher order linear ODE with constant coefficients and solution of second order ODE by the method of variation of parameters – Cauchy’s and Legendre’s linear equations - Simultaneous first order linear equations with constant coefficients.

PDEs:                                                                                                                         (6 Hours)

Cauchy-Euler equations, Solutions about ordinary points, Solutions about singular points. Method of Frobenius, Charpit’s method. Second solutions and Logarithm terms.Some mathematical models of PDEs, Fourier series solutions, Method of separation of variables, The D’Alembert solution.






Applications of Differential Equations:                                                              (4 Hours)

Formulation and solution of DEs related to Orthogonal Trajectories, Newton's Law of Cooling, simple harmonic Motion, One-Dimensional Conduction of Heat, mechanical & electrical oscillatory circuits &Chemical Problems.


At least a question MUST be set from each unit. Seven questions MUST be set from which five questions are attempted by the students.



·        Lectures

·        Reading assignments

·        Practical assignments


·        Whiteboard and Markers

·        Flip Charts