Kerala University B.Tech First Year (Combined First and Second Semester) Mathematics sylabbus

08.103 ENGINEERING MATHEMATICS- I

Module-I Applications  of  differentiation:–  Definition  of  Hyperbolic  functions  and  their derivatives- Successive differentiation- Leibnitz’ Theorem(without proof)- Curvature-  Radius of curvature- centre of curvature-   Evolute ( Cartesian  ,polar and parametric forms)  Partial  differentiation  and  applications:-  Partial  derivatives-  Euler’s  theorem  on homogeneous  functions-  Total  derivatives-  Jacobians-  Errors  and  approximations-  Taylor’s  series  (one  and  two  variables)  - Maxima  and minima  of  functions  of  two variables - Lagrange’s method- Leibnitz rule on differentiation  under integral sign. Vector  differentiation  and  applications  :-  Scalar  and  vector  functions- differentiation of vector functions-Velocity and acceleration- Scalar and vector fields- OperatorÑ-  Gradient-  Physical  interpretation  of  gradient-  Directional  derivative- 

Divergence- Curl- Identities involving Ñ (no proof) - Irrotational and solenoidal  fields – Scalar potential. 

Module-II

Laplace  transforms:-    Transforms  of  elementary  functions  -  shifting  property- Inverse  transforms-  Transforms  of  derivatives  and  integrals-  Transform  functions multiplied   by  t and divided by  t  - Convolution  theorem(without proof)-Transforms of unit  step  function,  unit  impulse  function  and  periodic  functions-second  shifiting theorem- Solution of  ordinary  differential  equations with  constant  coefficients using Laplace transforms. Differential  Equations  and  Applications:-    Linear  differential  equations  with constant  coefficients-  Method  of  variation  of  parameters  -  Cauchy  and  Legendre  equations Simultaneous  linear  equations with  constant  coefficients- Application  to orthogonal trajectories (cartisian form only).

Module-III Matrices:-Rank  of  a  matrix-  Elementary  transformations-  Equivalent  matrices- Inverse of a matrix by gauss-Jordan method- Echelon form and normal form- Linear dependence and independence of vectors- Consistency- Solution of a system  linear equations-Non homogeneous and homogeneous equations- Eigen values and eigen vectors – Properties of eigen values and eigen vectors- Cayley Hamilton theorem(no proof)-  Diagonalisation-  Quadratic  forms-  Reduction  to  canonical  forms-Nature  of quadratic forms-Definiteness, rank, signature and index.