PHY 221: Multivariable Calculus

3 credits | Prerequisites: PHY 220

Course rationale

This course is a natural continuation of the calculus of a single variable. Most often in physics and applied sciences, we need to deal with functions that depend on multiple variables. Vectors are also inseparable in physical applications, which is the natural language in many branches of physics and engineering. In this course, the students will learn the theory and applications of multivariable and vector calculus.

Course content

Vector-valued Functions of a Single Variable: derivatives, and integrals of vector valued functions; Differential Geometry of Vector-valued Function: tangent lines to vector-valued functions, arc length; curvature of plane and space curves; Partial Differentiation: functions of several variables, limits and continuity, partial derivatives; Directional Derivatives: gradient vectors and tangent planes; Extrema of Functions of Several Variables: Lagrange multipliers, Taylor’s formula; Vector Differential Calculus: vector fields, gradient, divergence, and curl; Line Integral: integral along a path; Multiple integrals: double and triple integrals, iterated integrals, areas and volumes, general multiple integrals; Change of Variables in Multiple Integrals: Jacobians, integrals in cylindrical and spherical coordinates; Green’s Theorem: proof and applications; Gauss’s Theorem: proof and applications; Stokes’ Theorem: proof and applications.

Course objectives

  1. Analyze and understand the function of multiple variables.
  2. Use vector analysis in problems of geometry and physics.
  3. Apply the techniques to evaluate the extrema of a function of several variables.
  4. Use vector integral calculus to compute different quantities of interest in mathematics and physics.
  5. Understand and apply different theorems of vector integral calculus


  1. Multivariable Calculus (9th edition) by James Stewart, Daniel K. Clegg, and Saleem Watson
  2. Vector Calculus (4th edition) by Susan Colley
  3. Calculus: Multivariable (9th edition) by Howard Anton, Irl C. Bivens, and Stephen Davis
  4. Calculus (12th edition) by Ron Larson, and Bruce H. Edwards
  5. Calculus (Volume 3) by Edwin Herman and Gilbert Strang