General Course Information

1. Syllabus MATH 3153
2. Generic Course Objectives
3. Classroom Behaviour
4. Checklist of Topics Covered

Topic 1: Vector Algebra (Lang Ch.1)

1. Notes on linear spaces

Topic 2: Lines and Planes (Lang Ch.1)

1. Notes and Exercises on Lines in R^n
2. Notes and Exercises on  Planes

Topic 3 . Curves and Paths (Lang Ch. 2 §1)

1. Notes on the cross product.
2. Determinants and Orientation in R^2.
3. Unit Speed Reparametrization.  

Topic 4 Arclength and Curvature (Lang Ch. 2 §2)

1. Notes on curvature.  
2. Frenet Formulae

Topic 5. Functions of Several Variables (Lang Ch. 3)

1. Level Curves
2. Scalar Function of Several Variables – Partial Derivatives
3. The Gradient

Topic 6. The Derivative (Lang, Chapters 4 and 9.1)

1. Reformulations of the Notion of Derivative
2. Chain Rule for Curves.
3. Orthogonal Families
4. The Derivative of Vector Valued Functions
5. The Chain Rule for Vector Valued Functions.

Topic 7. Curve Integrals

1. Notes and Exercises on Line Integrals and Conservative Fields
2. Article: “How to determine if a vector field is conservative”

Topic 8. Higher Derivatives. (Lang Ch. 6 §1-3)

1. Review of Taylor’s Theorem in one dimension.
2. Higher Derivatives and Taylor’s Theorem

Topic 9. Optimization. (Lang Ch. 7.1 – 7.3 )

1. Unconstrained Optimization.
2. Exercises on optimization.

Topic 10. Constrained Optimization. (Lang Ch. 7.4 )

1. Optimization with Constraints

Topic 11. Double and Triple Integrals.(Lang Ch. 12 )

1. Review Calculus I Notes on Definite Integrals.
2. The Double Integral.
3. Iterated Integrals and Charge Density
4. The Mean Value Theorem for Integrals

Topic 12. The Change of Variable Formula. (Lang §11.2, §13.1-13.3 )

1. Change of Variables Theorem

Topic 13. Parametrized Surfaces. (Lang Ch. 15.1-15.3 )

1. Surface Integrals
2. Green’s Theorem

Topic 15. Stokes’s Type Theorems (Lang Ch. 15 )

1. Physical Applications of Stokes’ Theorem (Reading for Physicists)

Practice Questions for Exams

1. MATH 3153 Exam I – Practice
2. MATH 3153 Exam II – Practice
3. MATH 3153 Exam III – Practice

Old Exams with Solutions

1. Exam I
2. Exam II
3. Exam III
4. Exam IV

Practice Problems with Solutions

1. Vector Geometry
2. Parametrized Curves
3. Derivative Problems
4. Unconstrained Optimization
5. Lagrange Multipliers
6. Integration
7. Line Integrals
8. Surface Integrals
9. Green’s Theorem
10. Divergence Theorem
11. Stokes Theorem

Related Notes

1. Determinants and Orientation in R^2
2. The Vector Product
3. Matrix Multiplication – Brief Review
4. Matrices and Linear Maps.
5. The Derivative of a Bilinear Map
6. Formulae for Integration with Differential Forms
7. Vector Area Form (For Elasticity Theory)