Engineering Mathematics Ii
Course Content and Outline
DETAILED COURSE CONTENT
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 orthogonalization process – Inner product spaces –Definition – Examples – Inequalities; Schwartz, Triangle (No proof). Decomposition of a vector with respect to a subspace and its orthogonal complement – Pythagorus 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 series‐C+iS' method.
Unit III: Gradient, Divergence, and Curl (6 Hours)
Gradient, Divergence, and Curl – Directional derivative – Irrotational and solenoid 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.
Definition of a Differential Equation, Formation of a Differential Equation, Ordinary and Partial Differential Equations, Order and Degree of a Differential Equation.
The 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. The concept of Particular and General Solution, Solving by Substitution
Linear Differential Equations: (7Hours)
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
MODE OF DELIVERY
Whiteboard and Markers