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# Applied Mathematics 4 Semester 4 (SE Second Year) BE Electronics Engineering University of Mumbai Topics and Syllabus

CBCGS [2017 - current]
CBGS [2013 - 2016]
Old [2000 - 2012]

## Topics with syllabus and resources

100.00 Module 1
101.00 Calculus of Variation
• Euler Langrange equation.
• Solution of Euler‘s Langrange equation (only results for different cases for function) independent of a variable, independent of another variable, independent of differentiation of a variable and independent of both variables.
• Isoperimetric problems, several dependent variables.
102.00 Functions Involving Higher Order Derivatives
• Rayleigh-Ritz method
200.00 Module 2
201.00 Linear Algebra: Vector Spaces
202.00 Vectors in N-dimensional Vector Space
• Properties
• Dot product
• Cross product
• Norm and distance properties in n-dimensional vector space.
• Metric spaces, vector spaces over real field
• Properties of vector spaces over real field, subspaces.
• Norms and normed vector spaces.
• Inner products and inner product spaces
• The Cauchy-Schwarz inequality, orthogonal Subspaces
• Gram-Schmidt Process.
300.00 Module 3
301.00 Linear Algebra: Matrix Theory
• Characteristic equation
• Eigenvalues and Eigenvectors
• Properties of Eigenvalues and Eigenvectors
• Cayley-Hamilton theorem, examples based on verification of CayleyHamilton theorem.
• Similarity of matrices
• Diagonalisation of matrix Functions of square matrix
• Derogatory and non-derogatory matrices Quadratic forms over real field
• Reduction of quadratic form to a diagonal canonical form
• rank, index, signature of quadratic form
• Sylvester‘s law of inertia
• value-class of a quadratic form of definite, semidefinite and indefinite Singular Value Decomposition.
400.00 Module 4
401.00 Complex Integration
• Line Integral
• Cauchy‘s Integral theorem for simply connected regions
• Cauchy‘s Integral formula
• Taylor‘s and Laurent‘s series.
402.00 Complex Variables
• Zeros
• Singularities
• Poles of f(z)
• Residues
• Cauchy‘s Residue theorem
• Applications of Residue theorem to evaluate real Integrals of different types.