2018 — MAT B41
The course syllabus is available here. All students in MAT B41 must print a paper copy of the syllabus and read it.
The suggested exercises for the first part of the course are listed in the syllabus. The suggested exercises for the second part of the course are available here.
Homework:
 Homework #1 (tex)
 Homework #2 (tex)
 Homework #3 (tex)
 Homework #4 (tex)
 Homework #5 (tex)
 Homework #6 (tex)
Evaluations:
Notes:
Course Services:
Course Outline
 Review: linear algebra and functions of several variables
 Differentiation of functions
 Multivariate Chain Rule
 Quadratic forms, Hessian matrices, and extrema of multivariate functions
 Constrained optimization and Lagrange multipliers
 Approximation by polynomials
 Integration of functions
 Change of Coordinates: spherical and polar coordinates
Students will develop familiarity with: “partial derivatives, gradients, tangent planes, the Jacobian matrix and chain rule, Taylor series, extremal problems, extremal problems with constraints and Lagrange multipliers, multiple integrals, spherical and cylindrical coordinates, law of transformation of variables.”
Prerequisites: This course requires Linear Algebra (MATA22H3 or MATA23H3 or MAT223H) and Calculus (MATA36H3 or MATA37H3 or MAT137Y or MAT157Y). This course will be very difficult if you do not have the prerequisites. If you want to attempt the course without these courses, please contact Parker immediately.
Note that this course excludes: MAT232H, MAT235Y, MAT237Y, and MAT257Y.
High level weekly overview:
 Weeks 13: Linear Algebra and Euclidean Space
 Weeks 36: Differentiation and Taylor Approximation
 Weeks 78: Optimization
 Weeks 910: Integration
 Weeks 1112: Change of Variables
Course Staff:
 Christopher Kennedy — christopherpa.kennedy@…
 David Pechersky — david.pechersky@…
 Kaidi Ye — kaidi.ye@…
 Xincheng Zhang — xincheng.zhang@…
 Xiucai Ding — xiucai.ding@…
To email your TA add “mail.utoronto.ca” to their address listed above.
Tutorial Schedule:
TUT3001  Wednesday 12:00  MW170  Christopher 
TUT3002  Thursday 12:00  SW 143  Kaidi 
TUT3003  Wednesday 15:00  IC 302  Christopher 
TUT3004  Tuesday 12:00  IC 230  Xiucai 
TUT3005  Wednesday 16:00  AC 334  David 
TUT3006  Friday 11:00  IC 300  Xiucai 
TUT3007  Thursday 15:00  IC 302  Kaidi 
TUT3008  Thursday 17:00  IC 326  Xincheng 
Weekly overview:

 Week 1 (notes)
 readings: 1.1 Vectors in Two and ThreeDimensional Space; 1.2 The Inner Product, Length, and Distance
 concepts:
 Coordinates
 Lines in two and three dimensions
 Planes in three or more dimensions
 Inner product
 Pythagorean theorem
 Dot product
 Euclidean length
 Orthogonality
 Week 2 (notes)
 readings: 1.3 Matrices, Determinants, and the Cross Product; Eric Moore’s course notes
 concepts:
 Matrix inverses
 Determinants
 Cofactor expansion
 Cross products
 Triangle inequality
 Week 3 (notes)
 readings: 1.5 nDimensional Euclidean Space; 2.2 Limits and Continuity
 concepts:
 Open sets
 Boundary points
 Limits
 Continuity
 Week 4 (notes)
 readings: 2.3 Differentiation; 2.5 Properties of Derivatives; 2.6 Gradients and Directional Derivatives
 concepts:
 Partial derivatives
 Directional derivatives
 Gradient vectors
 Composition of multivariate functions
 Multivariate chain rule
 Week 5 (notes)
 readings: 3.1 Iterated Partial Derivatives; 3.2 Taylor’s Theorem
 concepts:
 Second order partial derivatives
 Clairaut’s Theorem
 Taylor’s Theorem
 Approximation by polynomials
 Week 6
 Review, catchup, etc.
 Week 7
 Summary and Introduction to Optimization
 Week 8 (notes)
 readings: Course Notes: Quadratic forms and determinants; 3.3 Extrema of RealValued Functions
 concepts:
 Hessian matrix
 Critical points
 Minima and Maxima
 Week 9 (notes)
 readings: 3.4 Constrained Extrema and Lagrange Multipliers
 concepts:
 Constrained optimization
 Lagrange mutlipliers
 Week 10 (notes)
 readings: 5.1 Introduction to Double and Triple Integrals; 5.2 The Double Integral Over a Rectangle; 5.3 Double Integral Over More General Regions
 concepts:
 Volume and Area
 Cavalieri’s Principle
 Fubini’s Theorem
 Elementary Regions
 Week 11 (notes)
 readings: 5.4 Changing the Order of Integration; 5.5 The Triple Integral
 concepts:
 Simple Region
 Mean Value Theorem for Integrals
 Triple Integral
 Iterated Integral
 Week 12 (notes)
 readings: 6.1 Geometry of Maps from to ; 6.2 Change of Variables; 6.3 Applications (of Change of Variables)
 concepts:
 Mappings to .
 Onetoone mapping
 Onto mapping
 Change of Variables
 Polar Coordinates
 Cylindrical Coordinates
 Spherical Coordinates
 Jacobian Matrix