Math 3153 Calculus III

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2022-2023– Semester II

General Course Information

  1. Syllabus MATH 3153. (For Semester 2, 2022-23)
  2. Generic Course Objectives
  3. Classroom Behaviour
  4. On the use of electronic aparati
  5. In Praise of Lectures

We will be covering around 15 chapters of Lang (First Edition).  This means that you should read about one chapter every week.

Calendar

Week  Topics
1 Vector Algebra
2 Lines and Planes
3 Curves and paths in 2 and 3 dimensions
4 Arclength and Curvature
5 Functions of Several Variables
Exam I
6 The Derivative
7 Curve  Integrals 
8Higher Derivatives
9 Optimization 
10 Optimization with Constraints
Exam II
11 Double and Triple Integrals
12 The Change of Variable Formula
13 Green’s Theorem.   
Exam III
14 Parametrized Surfaces.
15 Stokes’  Type Theorems
Exam IVBased on the entire semester.

Sample Exams

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

Topic 1: Vector Algebra (Lang Ch.1)

  1. If necessary, rapidly review the foundational material in the videos below and following documents
  2. Read Chapter I of Lang1 and attempt as many exercises as time permits.
  3. Read class notes on linear spaces
  4. View Kahn Academy videos on vector spaces. 

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

  1. Read class notes:  Lines in R^n and attempt the exercises therein.
  2. Kahn Video on the Paramemtrization of Lines
  3. Read class notes:  Planes and attempt the exercises therein.
  4. Kahn video on Parametric Curves

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

  1. Read notes on the cross product.
  2. Read the article Determinants and Orientation in R^2.
  3. View the  Kahn videos on Vector Valued Functions.
  4. View articles on parametric curves  and applets  at Math Insight. 
  5. Read (Lang Chapter II Section 1) and do exercises in Section 2.1. 
  6. Exam I practice exercises.

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

  1. Read Lang Chapter 2,  Section 2.2 on arclength and curvature and do exercises therein.
  2.  Read Lang on paramtrization by arclength (Unit speed reparametrization).
  3. To reinforce  intuition see the applets and images  at Math Insight.
  4. View more  arclength examples at Math Insight.
  5. Read class notes on unit speed reparametrization.  
  6. Read class notes on curvature.  
  7. View these 4 Kahn Academy  videos on the derivative and  curvature.
  8.  Continue with Exam I practice exercises.

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

  1. View Kahn videos on Kahn videos on the Partial Derivative and Gradient.
  2. Read Lang Chapter 3, and do exercises therein.
  3. Read: Reformulation of the 1-dimensional derivative

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

  1. Read: notes on the derivative of a scalar function, directional derivatives and the gradient.
  2.  View Kahn videos on the Partial Derivative and Gradient.
  3. Read: Chain Rule for Curves.
  4. Do exercises (see pages 11 and 12) on the Derivative.
  5. Optional Reading on the Matrices and Linear Maps.
  6. The chain rule for vector valued functions.

Topic 7. Curve Integrals

  1. View Kahn Videos on Line Integrals and Conservative Fields.
  2. Read class notes: Line Integrals and potential functions and do exercises therein.
  3. Read online article “How to determine if a vector field is conservative”.

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

  1. Read Chapter 6 of Lang on Higher Derivatives and Taylor’s Theorem and attempt as many exercises as time permits.
  2. Review Taylor’s Theorem in one dimension.

Topic 9. Optimization.

  1. Watch this sequence of Kahn Vdeos on Optimzation.
  2. Read Sections 7.1 – 7.3, of Lang on Optimization.
  3. Read class notes on optimization.
  4. Do exercises on optimization.

Topic 10. Constrained Optimization.

  1. Read Chapter 7.4 of Lang on Lagrange Multipliers.
  2. Watch this sequence of Kahn Vdeos on Constrained Optimzation.
  3. Read class notes on Lagrange Multipliers.
  4. Read notes at Kahn Academy.

Topic 11. Double and Triple Integrals.

  1. Read § 12.1–12.2 of Lang on Double Integrals.
  2. Watch this sequence of Kahn Vdeos on Double Integration.
  3. Read Calculus I notes on Definite Integrals.
  4. Read notes on the Double Integral.
  5. Read notes on Iterated Integrals and Charge Density.

Topic 12. The Change of Variable Formula.

  1. View the 5 Kahn videos on Jacobians.
  2. Read §11.2 of Lang on the Jacobian.
  3. Read §13.1-13.3 of Lang on the Change of variable formula.
  4. Read notes on the change of variables formula.

Topic 13. Parametrized Surfaces.

  1. View these three Kahn Academy videos on parametrizing a surface.
  2. View Kahn Academy lesson and videos on surface integrals.
  3. Read §15.1, §15.2, §15.3 of Lang on Surface Integrals and attempt as many exercises as time permits.
  4. Read Class notes on surface integrals.
  1. Read §14.1, §14.2 of Lang on Green’s Theorem and
  2. Exercises:

Topic 15. Stokes’s Type Theorems

  1. Read §15.4, §15.5, §15.6 of Lang on the Divergence Theorem and Stoke’s Theorem
  2. Preparation for Exam IV.

Note: Exam IV will be based on the entire semester and is worth 200 points.

Miscellaneous Notes

  1. Determinants and Orientation in R^2
  2. The Vector Product
  3. Matrix Multiplication -Brief Review
  4. The Derivative
  5. The Chain Rule for Curves
  6. The Chain Rule for Vector Valued Functions
  7. Unit Speed Reparametrization
  8. Frenet Formulae
  9. Line Integrals and Conservative Fields
  10. Optimization – Summary
  11. Optimization with Constraints
  12. Double Integrals
  13. Change of Variables Theorem
  14. Iterated Integrals and Charge Density
  15. Surface Integrals
  16. Formulae for Integration
  17. Physical Applications of Stokes’ Theorem
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