# Advanced Mathematical Techniques in Chemical Engineering

IIT Kharagpur Course , Prof. S. De

### Lecture 1: Introduction to vector space

##### Lecture Details :

Advanced Mathematical Techniques in Chemical Engineering by Prof. S. De, Department of Chemical Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in

##### Course Description :

Introduction of vector space;Metric, Norm, Inner Product space;Examples
Onto, into, one to one function, completeness of space
Vectors: Linear combination of vectors, dependent/independent vectors; Orthogonal and orthonormal vectors; Gram-Schmidt orthogonalization; Examples
Contraction Mapping: Definition; Applications in Chemical Engineering; Examples
Matrix, determinants and properties
Eigenvalue Problem:Various theorems; Solution of a set of algebraic equations; Solution of a set of ordinary differential equations; Solution of a set of non-homogeneous first order ordinary differential equations (IVPs)
Applications of eigenvalue problems: Stability analysis; Bifurcation theory; Examples
Partial Differential equations:Classification of equations; Boundary conditions;Principle of Linear superposition
Special ODEs and Adjoint operators:Properties of adjoint operator; Theorem for eigenvalues and eigenfunctions;
Solution of linear, homogeneous PDEs by separation of variables: Cartesian coordinate system & different classes of PDEs; Cylindrical coordinate system ; Spherical Coordinate system
Solution of non-homogeneous PDEs by Green's theorem
Solution of PDEs by Similarity solution method
Solution of PDEs by Integral method
Solution of PDEs by Laplace transformation
Solution of PDEs by Fourier transformation

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