Multivariate Calculus by Ravendra Kumar and Dr. Sooraj Pal Singh is a comprehensive and concept-driven textbook designed according to the FYUGP/NEP syllabus for B.Sc. Mathematics Honours 4th and 5th Semesters. Published by Mahaveer Publications, this book offers a strong foundation in the differential and integral calculus of several variables—one of the most essential components of higher mathematics.

The book introduces students to the concepts of partial differentiation, multiple integrals, vector calculus, Jacobians, scalar and vector fields, directional derivatives, gradient, divergence, curl, line and surface integrals, and integral theorems. Written in a clear and student-friendly manner, it balances rigorous mathematical theory with illustrative examples, solved problems, and exercise sets.

The content provides not only the conceptual depth required for undergraduate studies but also connects multivariable calculus to real-world applications in physics, engineering, optimization, data science, and applied mathematics.

Chapter 1: Function of Several Variables

• Concept of multivariable functions

• Domain, range and graph of functions of two or more variables

• Limits and continuity of multivariable functions

• Partial derivatives and directional derivatives

• Total derivative

• Higher order partial derivatives

• Homogeneous functions and Euler’s theorem

• Taylor’s theorem for two variables

• Maxima and minima of functions of several variables

• Lagrange’s method of multipliers

Chapter 2: Double Integration over Rectangular & Non-Rectangular Regions

• Introduction to double integrals

• Evaluation of double integrals over rectangular regions

• Evaluation over non-rectangular regions

• Geometrical interpretation and area evaluation using double integrals

• Changing order of integration

• Applications to mass, centroid & surface area

Chapter 3: Triple Integrals

• Introduction to triple integrals

• Evaluation of triple integrals

• Triple integrals in rectangular, cylindrical & spherical coordinates

• Applications to volume and physical quantities

• Solid regions in 3-D geometry

Chapter 4: Change of Variables in Double & Triple Integration

• Transformation of variables

• Jacobians and their properties

• Change of variables in double integration

• Change of variables in triple integration

• Applications in coordinate transformations

Chapter 5: Stoke’s Theorem, Green’s Theorem & Divergence Theorem

Green’s Theorem

• Line integral and region in the plane

• Green’s theorem in the plane

• Applications to area evaluation

• Physical interpretations

Stoke’s Theorem

• Surface integral and orientation

• Curl of a vector field

• Statement and verification of Stokes’ theorem

• Applications in fluid flow & electromagnetics

Divergence (Gauss) Theorem

• Divergence of a vector field

• Surface and volume integrals

• Statement and proof outline

• Applications in flux calculations