Welcome to Multi-Variable and Vector Calculus. This is a really exciting subject and is the first step into allowing one to tackle real world physics and engineering problems.

Sections marked with an asterisk * should be regarded as optional, and everything else as essential. The ones chosen to be marked with an asterisk are ones which are normally left out of multivariable calculus courses, but may be included in an honours course. One of the goals is to be able to cater to as wide of an audience as possible, and for this reason those sections were written.

This is a course on calculus, not analysis. The goal is for students to understand how to compute important quantities and objects in multivariable calculus, and to understand the concepts without proof and be able to interpret concepts geometrically.

Introduction - What is Multivariable Calculus?

1. Working knowledge of single variable differential calculus (Calculus I)
2. Working knowledge of single variable integral calculus (Calculus II)
3. Rudimentary knowledge of linear algebra; any first course in linear algebra is more than sufficient
4. Knowledge of ordinary differential equations may be useful
5. An exposure to introductory physics will be useful for developing physical intuition

• Unit 1 - Introduction to 3D Space and Vectors
1. Three Dimensional Cartesian Coordinates - Exercises
2. Vectors and Their Geometry - Exercises
3. Algebraic Operations with-Vectors - Exercises
4. Norm - Exercises
5. Projections - Exercises
6. Dot Product - Exercises
7. Determinant and Cross Product - Exercises
8. Equations of Lines - Exercises
9. Equations of Planes - Exercises
10. Quadric Surfaces - Exercises
11. Cylindrical Coordinates - Exercises
12. Spherical Coordinates - Exercises
13. General Surfaces - Exercises
• Unit 2 - Space Curves
1. Vector Functions - Exercises
2. Limits and Continuity - Exercises
3. Differentiation and Integration of Curves - Exercises
4. Arc Length - Exercises
5. Curvature and Torsion - Exercises
6. Frenet Serret Equations - Exercises
• Unit 3 - Partial Differentiation
1. Functions of Multiple Variables - Exercises
2. Contour Plots and Level Sets - Exercises
3. Limits and Continuity - Exercises
4. Partial Differentiation - Exercises
5. Tangent Planes and Linear Approximations - Exercises
6. Gradient Vector - Exercises
7. The Differential - Exercises
8. Non Independent Variables - Exercises
9. Chain Rule - Exercises
10. Differentiation of Integrals - Exercises
11. Directional Derivative - Exercises
12. Extrema 1 - Exercises
13. Extrema 2 - Exercises
14. Lagrange Multipliers 1 - Exercises
15. Lagrange Multipliers 2 - Exercises
16. Bordered Hessian - Exercises
17. Lagrange Remainder - Exercises
18. The Jacobian Derivative - Exercises
19. Higher Dimensional Lagrange Multipliers - Exercises
20. Inflection and Saddle Points Lagrange Multipliers - Exercises
• Unit 4 - Multiple Integration
1. Riemann Sums - Exercises
2. Double Integral Over a Rectangle - Exercises
3. Double Integral Over a General Plane Region - Exercises
4. Double Integral in Polar Coordinates - Exercises
5. Surface Area - Exercises
6. Triple Integrals - Exercises
7. Triples Integrals In Cylindrical Coordinates - Exercises
8. Triple Integrals in Spherical Coordinates - Exercises
9. Change of Variables - Exercises
10. Applications - Exercises
• Unit 5 - Vector Calculus
1. Vector Fields - Exercises
2. Line Integrals in 2D - Exercises
3. Fundamental Theorem of Line Integrals - Exercises
4. Green's Theorem - Exercises
5. Divergence - Exercises
6. Curl - Exercises
7. Laplacian - Exercises
8. Surface Integrals - Exercises
9. Stokes Theorem - Exercises
10. Divergence Theorem - Exercises
11. Applications - Exercises
12. Flow of a Vector Field - Exercises
13. Maxwell's Equations - Exercises
14. Differential Forms - Exercises
15. Green' Identities - Exercises
• Supplementary Material
1. Chain Rules - Exercises
2. Coordinate Change-Tensor - Exercises
3. Coordinate Change - Exercises
4. Curvilinear Coordinates - Exercises
5. Notation - Exercises
6. Product Rules - Exercises

Sources

1. Calculus, 7 ed. - Stewart, James (2012)
2. Vector Analysis - Spiegel, Murray R. (1959)
3. Differential Geometry of Curves and Surfaces - 2 ed. do Carmo, Manfredo (2016)
4. Calculus on Manifolds - Spivak, Michael (1965)