|| 10 MAT-11
|| Engineering Maths-I
UNIT – 1
Differential Calculus - 1
Determination of nth derivative of standard functions-illustrative examples*.
Leibnitz’s theorem (without proof) and problems.
Rolle’s Theorem – Geometrical interpretation. Lagrange’s and Cauchy’s
mean value theorems. Taylor’s and Maclaurin’s series expansions of function
of one variable (without proof).
UNIT – 2
Differential Calculus - 2
Indeterminate forms – L’Hospital’s rule (without proof), Polar curves: Angle
between polar curves, Pedal equation for polar curves. Derivative of arc
length – concept and formulae without proof. Radius of curvature - Cartesian,
parametric, polar and pedal forms.
UNIT – 3
Differential Calculus - 3
Partial differentiation: Partial derivatives, total derivative and chain rule,
Taylor’s expansion of a function of two variables-illustrative examples*.
Maxima and Minima for function of two variables. Applications – Errors and
UNIT – 4
Scalar and vector point functions – Gradient, Divergence, Curl, Laplacian,
Solenoidal and Irrotational vectors.
Vector Identities: div (øA), Curl (øA) Curl (grad ø ) div (CurlA) div (A x B )
& Curl (Curl A) .
Orthogonal Curvilinear Coordinates – Definition, unit vectors, scale factors,
orthogonality of Cylindrical and Spherical Systems. Expression for Gradient,
Divergence, Curl, Laplacian in an orthogonal system and also in Cartesian,
Cylindrical and Spherical System as particular cases – No problems
UNIT – V
Differentiation under the integral sign – simple problems with constant
limits. Reduction formulae for the integrals of
sinn x, cosn x, s i n m x c o s n x and evaluation of these integrals with
standard limits - Problems.
Tracing of curves in Cartesian, Parametric and polar forms – illustrative
examples*. Applications – Area, Perimeter, surface area and volume.
Computation of these in respect of the curves – (i) Astroid:
2 2 2
x 3+y 3 =a 3
(ii) Cycloid: x =a (q -sinq ), y =a (1 - cosq ) and (iii) Cardioid:
r =a (1+ cosq )
UNIT – VI
Solution of first order and first degree equations: Recapitulation of the
method of separation of variables with illustrative examples*. Homogeneous,
Exact, Linear equations and reducible to these forms. Applications -
UNIT – VII
Recapitulation of Matrix theory. Elementary transformations, Reduction of
the given matrix to echelon and normal forms, Rank of a matrix, consistency
of a system of linear equations and solution. Solution of a system of linear
homogeneous equations (trivial and non-trivial solutions). Solution of a
system of non-homogeneous equations by Gauss elimination and Gauss –
UNIT – VIII:
Linear Algebra -2
Linear transformations, Eigen values and eigen vectors of a square matrix,
Similarity of matrices, Reduction to diagonal form, Quadratic forms,
Reduction of quadratic form into canonical form, Nature of quadratic forms