This comprehensive and self-contained text provides a thorough understanding of the concepts and applications of Linear algebra and Vector Calculus. It is written in such a manner that beginners can develop an interest in the subject. Besides providing the essentials of the theory, the book helps in developing problem-solving techniques and sharpens the skill of logical thinking.

The book covers a wide range of topics such as Matrices, vector Algebra, Vector calculus, Multiple Integrals, Applications of differential equations. It too serves as a textbook for the students of under graduation, post-graduation and Engineering.

Each chapter has been planned as an independent unit. The exercise questions given at the end of each chapter are also useful for the competitive examinations like GATE, IES, ICS.

**Key Features**

Each chapter is saturated with much needed texts, and suitable illustrations.

The language is simple and precise.

Complete proofs/verifications provided wherever necessary.

Exercises with hints and answers are given at the end of each chapter.

**Dr. Bhavanari Satyanarayana,** Professor, Department of Mathematics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India.

**Dr. T.V. Pradeep Kumar,** Assistant Professor, Department of Mathematics, University College of Engineering, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India.

**Mr. D. Srinivasulu,** Assistant Professor, Department of Mathematics, NRI College of Engineering, Guntur, Andhra Pradesh, India.

**1 Matrices and Linear System of Equations**

1.1 Matrices

1.2 Algebra of Matrices

1.3 Rank of a Matrix

1.3.1 Solved Problems

1.4 Normal Form of a Matrix

1.4.1 Solved Problems

1.5 Linear System of Equations

1.6 Solution of System of Linear Equations

1.6.1 Solved Problems

1.7 Applications

1.8 Solved Problems

1.9 Exercise

**2 Eigenvalues, Eigenvectors and Quadratic Forms**

2.1 Eigenvalues and Eigenvectors

2.2 Properties of Eigenvalues

2.3 Algebraic and Geometric Multiplicity of an Eigen value

2.3.1 Solved Problems

2.4 Polynomial Matrix

2.5 Cayley–Hamilton Theorem

2.5.1 Solved Problems

2.6 Matrix Diagonalization

2.6.1 Solved Problems

2.7 Quadratic Form

2.8 Canonical Form: (principal axis form or sum of squares form)

2.8.1 Solved Problems

2.9 Exercise

**3 Multiple Integrals**

3.1 Definite Integrals

3.1.1 Standard Integrals

3.1.2 Reduction Formula

3.2 Double Integrals

3.2.1 Properties of Double Integration

3.2.2 Evaluation of Double Integrals

3.2.3 Solved Problems

3.2.4 Change of the Order of Integration

3.2.5 Solved Problems

3.2.6 Evaluation of Double Integrals in Polar Coordinates

3.2.7 Solved Problems

3.3 Triple Integrals

3.3.1 Cylindrical Co-ordinates

3.3.2 Solved Problems

3.4 Change of Variable

3.4.1 Solved Problems

3.5 Solved Problems

3.6 Exercise

**4 Special Function**

4.1 Beta and Gamma Functions

4.2 Properties of Beta Function

4.3 Transformation of Gamma Function (Different Forms of Gamma Function)

4.3.1 Solved Problems

4.4 Solved Problems

4.5 Exercise

**5 Vector Differentiation**

5.1 Vector Differentiation

5.1.1 Solved Problems

5.1.2 Solved Problems on Directional Derivative of a Scalar Point Function

5.2 Divergence of Vector

5.2.1 Solved Problems

5.3 Curl of a Vector

5.3.1 Solved Problems

5.4 Operators

5.4.1 Solved Problems

5.5 Solved Problems

5.6 Exercise

**6 Vector Integration**

6.1 Integrals

6.1.1 Line Integrals

6.1.2 Surface Integrals

6.1.3 Volume Integrals

6.1.4 Solved Problems

6.2 Greens Theorem

6.2.1 Solved Problems

6.3 Stoke’s Theorem

6.4 Gauss Divergence Theorem

6.5 Coordinates

6.5.1 Curvilinear Coordinates

6.5.2 Cylindrical Coordinates

6.5.3 Solved Problems

6.6 Solved Problems

**7 Curve Tracing**

7.1 Curve Tracing in Cartesian Form

7.1.1 Solved Problems

7.2 Curve Tracing Polar Form

Solved Problems

7.3 Curve Tracing Parametric Form

7.1.1 Solved Problems

7.4 Some Special Cases

7.5 Exercise

**8 Some Applications of Multiple Integrals**

8.1 Formulas Related to Mass of Plane, Lamina/ Solid

8.1.1 Mass of Plane Lamina

8.1.2 Mass of a Solid

8.1.3 Centre of Mass of Plane Lamina (or) Centre of Gravity

8.1.4 Centre of Mass (or) Centre of Gravity of Solid

8.2 Solved Problems

8.3 Exercise

**Answers**

Index