Engineering Mathematics Ii
Course Content and Outline
Unit I: Linear Algebra
Eigenvalue problem (6 Hours )
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)
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.
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 series‐C+iS' method.
Unit III: Gradient,
Divergence and Curl (6 Hours)
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
of a Differential Equation, Formation of a Differential Equation, Ordinary and
Partial Differential Equations, Order and Degree of a Differential Equation.
of first Order and first Degree: (8 Hours)
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
Differential Equations: (6 Hours)
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
PDEs: (6 Hours)
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)
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
least a question MUST be set from each unit. Seven questions MUST be set from
which five questions are attempted by the students.
MODE OF DELIVERY
Whiteboard and Markers