MA6151 MATHEMATICS - I

L T P C

3 1 0 4

OBJECTIVES:

• To develop the use of matrix algebra techniques this is needed by engineers for practical applications.

• To make the student knowledgeable in the area of infinite series and their convergence so that he/ she will be familiar with limitations of using infinite series approximations for solutions arising in mathematical modeling.

• To familiarize the student with functions of several variables. This is needed in many branches of engineering.

• To introduce the concepts of improper integrals, Gamma, Beta and Error functions which are needed in engineering applications.

• To acquaint the student with mathematical tools needed in evaluating multiple integrals and their usage.

UNIT I MATRICES

Eigenvalues and Eigenvectors of a real matrix - Characteristic equation - Properties of eigenvalues and eigenvectors - Statement and applications of Cayley-Hamilton Theorem - Diagonalization of matrices - Reduction of a quadratic form to canonical form by orthogonal transformation - Nature of quadratic forms.

UNIT II SEQUENCES AND SERIES 9+3

Sequences: Definition and examples - Series: Types and Convergence - Series of positive terms - Tests of convergence: Comparison test, Integral test and D'Alembert's ratio test - Alternating series - Leibnitz's test - Series of positive and negative terms - Absolute and conditional convergence.

UNIT III APPLICATIONS OF DIFFERENTIAL CALCULUS 9+3

Curvature in Cartesian co-ordinates - Centre and radius of curvature - Circle of curvature - Evolutes

- Envelopes - Evolute as envelope of normals.

UNIT IV DIFFERENTIAL CALCULUS OF SEVERAL VARIABLES 9+3

Limits and Continuity - Partial derivatives - Total derivative - Differentiation of implicit functions - Jacobian and properties - Taylor's series for functions of two variables - Maxima and minima of functions of two variables - Lagrange's method of undetermined multipliers.

UNIT V MULTIPLE INTEGRALS 9+3

Double integrals in cartesian and polar coordinates - Change of order of integration - Area enclosed by plane curves - Change of variables in double integrals - Area of a curved surface - Triple integrals

- Volume of Solids.

TOTAL: 60 PERIODS

TEXT BOOKS:

* Bali N. P and Manish Goyal, "A Text book of Engineering Mathematics", Eighth Edition, Laxmi Publications Pvt Ltd., (2011).

* Grewal. B.S, "Higher Engineering Mathematics", 41 Edition, Khanna Publications, Delhi, (2011).

REFERENCES:

* Dass, H.K., and Er. Rajnish Verma," Higher Engineering Mathematics", S. Chand Private Ltd., (2011).

* Glyn James, "Advanced Modern Engineering Mathematics", 3rd Edition, Pearson Education, (2012).

* Peter V. O'Neil," Advanced Engineering Mathematics", 7th Edition, Cengage learning, (2012).

* Ramana B.V, "Higher Engineering Mathematics", Tata McGraw Hill Publishing Company, New Delhi, (2008).

L T P C

3 1 0 4

OBJECTIVES:

• To develop the use of matrix algebra techniques this is needed by engineers for practical applications.

• To make the student knowledgeable in the area of infinite series and their convergence so that he/ she will be familiar with limitations of using infinite series approximations for solutions arising in mathematical modeling.

• To familiarize the student with functions of several variables. This is needed in many branches of engineering.

• To introduce the concepts of improper integrals, Gamma, Beta and Error functions which are needed in engineering applications.

• To acquaint the student with mathematical tools needed in evaluating multiple integrals and their usage.

UNIT I MATRICES

Eigenvalues and Eigenvectors of a real matrix - Characteristic equation - Properties of eigenvalues and eigenvectors - Statement and applications of Cayley-Hamilton Theorem - Diagonalization of matrices - Reduction of a quadratic form to canonical form by orthogonal transformation - Nature of quadratic forms.

UNIT II SEQUENCES AND SERIES 9+3

Sequences: Definition and examples - Series: Types and Convergence - Series of positive terms - Tests of convergence: Comparison test, Integral test and D'Alembert's ratio test - Alternating series - Leibnitz's test - Series of positive and negative terms - Absolute and conditional convergence.

UNIT III APPLICATIONS OF DIFFERENTIAL CALCULUS 9+3

Curvature in Cartesian co-ordinates - Centre and radius of curvature - Circle of curvature - Evolutes

- Envelopes - Evolute as envelope of normals.

UNIT IV DIFFERENTIAL CALCULUS OF SEVERAL VARIABLES 9+3

Limits and Continuity - Partial derivatives - Total derivative - Differentiation of implicit functions - Jacobian and properties - Taylor's series for functions of two variables - Maxima and minima of functions of two variables - Lagrange's method of undetermined multipliers.

UNIT V MULTIPLE INTEGRALS 9+3

Double integrals in cartesian and polar coordinates - Change of order of integration - Area enclosed by plane curves - Change of variables in double integrals - Area of a curved surface - Triple integrals

- Volume of Solids.

TOTAL: 60 PERIODS

TEXT BOOKS:

* Bali N. P and Manish Goyal, "A Text book of Engineering Mathematics", Eighth Edition, Laxmi Publications Pvt Ltd., (2011).

* Grewal. B.S, "Higher Engineering Mathematics", 41 Edition, Khanna Publications, Delhi, (2011).

REFERENCES:

* Dass, H.K., and Er. Rajnish Verma," Higher Engineering Mathematics", S. Chand Private Ltd., (2011).

* Glyn James, "Advanced Modern Engineering Mathematics", 3rd Edition, Pearson Education, (2012).

* Peter V. O'Neil," Advanced Engineering Mathematics", 7th Edition, Cengage learning, (2012).

* Ramana B.V, "Higher Engineering Mathematics", Tata McGraw Hill Publishing Company, New Delhi, (2008).

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