Vtu syllabus

ENGINEERING MATHEMATICS – I

Sub Code: 10MAT11
IA Marks: 25
Hrs/ Week: 04 Exam Hours: 03
Total Hrs: 52 Exam Marks: 100

PART-A

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). 6 Hours

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. 7 Hours

UNIT – 3: Differential Calculus - 3

Partial differentiation: Partial derivatives, total derivative and chain rule, Jacobians-direct evaluation.
Taylor’s expansion of a function of two variables-illustrative examples*. Maxima and Minima for function of two variables. Applications – Errors and approximations. 6 Hours

UNIT – 4: Vector Calculus

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. 7 Hours

PART-B

UNIT – V: Integral Calculus

Differentiation under the integral sign – simple problems with constant limits. Reduction formulae for the integrals of sinn x, cosn x, s i nm 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 ) 6 Hours

UNIT – VI: Differential Equations

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 -orthogonal trajectories. 7 Hours

UNIT – VII: Linear Algebra-1

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 – Jordan methods. 6 Hours

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. 7 Hours

Note: * In the case of illustrative examples, questions are not to be set. 7

Text Books:

1. B.S. Grewal, Higher Engineering Mathematics, Latest edition, Khanna Publishers
2. Erwin Kreyszig, Advanced Engineering Mathematics, Latest edition, Wiley Publications.

Reference Books:

1. B.V. Ramana, Higher Engineering Mathematics, Latest edition, Tata Mc. Graw Hill Publications.
2. Peter V. O’Neil, Engineering Mathematics, CENGAGE Learning India Pvt Ltd.Publishers

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All the Best ..!!