Applied Mathematics I (REVISED 2007)



 1.1        Module 1
            Complex numbers.
            1.1.1     Review of complex numbers. Cartesian, Polar and                      02
                     Exponential form of a complex number.
            1.1.2     De Moiver’s Theorem (without proof). Powers and roots                03
                     of Exponential and Trigonometric functions.
            1.1.3     Circular and Hyperbolic functions.

1.2        Module 2
            Complex numbers and successive differentiation.
            1.2.1    Inverse circular and Inverse Hyperbolic Functions                     03
                    Logarithmic functions
            1.2.2     Separation of real and imaginary parts of all types of               02
            1.2.3     Successive differentiation –nth derivative of standard               04
                                  ax         -1         m          -m
                      functions-e   , (ax=b)   ,  (ax=b)  , (ax=b)   , log (ax + b)
                     sin (ax + b) Cos (ax+b). eax  sin (bx+c). eax    cos (bx+c).

            1.2.4     Leibnitz’s theorem (without proot) and problems.                     03
1.3        Module 3
           Partial differentiation
            1.3.1  Partial derivatives of first and higher order, total differential
                   coefficients, total differentials, differentiation of composite         05
                   and implicit functions.
            1.3.2   Euler’s theorem on Homogeneous function with two and
                    three independent Variables (with proof), deductions from              03
                    Euler’s theorem.                                                       Total:08
1.4        Module 4
           Application of partial differentiation, Mean Value theorems
            1.4.1  Errors and approximations. Maxima and Minima of a
                   function of two independent variables. Lagrange’s method                04
                   of undetermined multipliers with one constraint.
            1.4.2   Rolle’s theorem, Lagrange’s mean value theorem,
                    Cauchy’s mean value theorem (all theorems without                      03
                    proof). Geometrical interpretation and problems.                       Total:07
1.5         Module 5
            Vector algebra & Vector calculus
            1.5.1    Vector triple product and product of four vectors.                         01
            1.5.2    Differentiation of a vector function of a single scalar
                     variable. Theorems on derivatives (without proof). curves                  02
                      in space concept of a tangent vector (without problems)
            1.5.3    Scalar point function and vector point function. Vector                    06
                    differential operator del. Gradient, Divergence and curl-
                    definitions, Properties and problems. Applications-Normal,                  Total:09
                    directional derivatives, Solenoidal and lrrotational fields.
1.6         Module 6
            Infinite series, Expansion of functions and indeterminate forms.
            1.6.1    Infinite series-Idea of convergence and divergence. D’                     02
            Alembert’s root test, Cauchy’s root test.
            1.6.2    Taylor’s theorem (Without proof) Taylor’s series and
                     Maclaurin’s series (without proof) Expansion of standard                   04
                     series such as e  , sinx, cosx, tanx, sinhx, coshx, tanhx,
                     log(1+x), sin-1x – tan-1x, binomial series, expansion of

                     functions in power series.
            1.6.3    Indeterminate forms-
             0  x                          x                                                    02
                                   0    0  x
               ,  ,0 x 8, 8 – 8, 0  , 8  ,1 BHospitalsrule – problem sin volvingseriesalso.
             –  x

Reference Books:
            • A textbook of Applied Mathematies. P.N. & J.N wartikar, volume
            1 & 2 pune Vidyarthi Griha.
            •Higher Engineering Mathematics Dr. B.S. Grewal, Khanna
            •Advanced Engineering Mathematics, Erwai Kreyszing, Wiley
            Eastern Limited, 8      Ed.
            •Vector analysis- Murray R., Spiegal- Scham series
            •Higher Engineering mathematics by B.V. Ramana-Tata McGraw