Engineering Mathematics -1 Notes
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UNIT – I
Sequences – Series
Basic definitions of Sequences and series – Convergences and divergence – Ratio test – Comparison test – Integral test – Cauchy’s root test – Raabe’s test – Absolute and conditional convergence
UNIT – II
Functions of Single Variable
Rolle’s Theorem – Lagrange’s Mean Value Theorem – Cauchy’s mean value Theorem – Generalized Mean Value theorem (all theorems without proof) Functions of several variables – Functional dependence- Jacobian- Maxima and Minima of functions of two variables with constraints and without constraints
UNIT – III
Application of Single variables
Radius, Centre and Circle of Curvature – Evolutes and Envelopes Curve tracing – Cartesian , polar and Parametric curves.
UNIT – IV
Integration & its applications
Riemann Sums , Integral Representation for lengths, Areas, Volumes and Surface areas in Cartesian and polar coordinates multiple integrals – double and triple integrals – change of order of integration- change of variable
UNIT – V
Differential equations of first order and their applications
Overview of differential equations- exact, linear and Bernoulli. Applications to Newton’s Law of cooling, Law of natural growth and decay, orthogonal trajectories and geometrical applications.
UNIT – VI
Higher Order Linear differential equations and their applications
Linear differential equations of second and higher order with constant coefficients, RHS term of the type f(X)= e ax , Sin ax, Cos ax, and xn, e ax V(x), x n V(x), method of variation of parameters. Applications bending of beams, Electrical circuits, simple harmonic motion.
UNIT – VII
Laplace transform and its applications to Ordinary differential equations
Laplace transform of standard functions – Inverse transform – first shifting Theorem, Transforms of derivatives and integrals – Unit step function – second shifting theorem – Dirac’s delta function – Convolution theorem – Periodic function – Differentiation and integration of transforms-Application of Laplace transforms to ordinary differential equations.
UNIT – VIII
Vector Calculus: Gradient- Divergence- Curl and their related properties Potential function – Laplacian and second order operators. Line integral – work done ––- Surface integrals – Flux of a vector valued function. Vector integrals theorems: Green’s -Stoke’s and Gauss’s Divergence Theorems (Statement & their Verification).
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