## 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:

Evaluations:

Notes:

Course Services:

Course Outline

• Review: linear algebra and functions of several variables
• Differentiation of functions $f: \mathbf{R}^n \rightarrow \mathbf{R}^k$
• Multivariate Chain Rule
• Quadratic forms, Hessian matrices, and extrema of multivariate functions
• Constrained optimization and Lagrange multipliers
• Approximation by polynomials
• Integration of functions $f: \mathbf{R}^n \rightarrow \mathbf{R}$
• 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 pre-requisites. 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 1-3: Linear Algebra and Euclidean Space
• Weeks 3-6: Differentiation and Taylor Approximation
• Weeks 7-8: Optimization
• Weeks 9-10: Integration
• Weeks 11-12: 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@…

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 Three-Dimensional 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
• Co-factor expansion
• Cross products
• Triangle inequality

• Week 3 (notes)
• readings: 1.5 n-Dimensional 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
• 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, catch-up, etc.
• Week 7
• Summary and Introduction to Optimization
• Week 8 (notes)
• readings: Course Notes: Quadratic forms and determinants; 3.3 Extrema of Real-Valued 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 $\mathbf{R}^2$ to $\mathbf{R}^2$; 6.2 Change of Variables; 6.3 Applications (of Change of Variables)
• concepts:

• Mappings $\mathbb{R}^2$ to $\mathbb{R}^2$.
• One-to-one mapping
• Onto mapping
• Change of Variables
• Polar Coordinates
• Cylindrical Coordinates
• Spherical Coordinates
• Jacobian Matrix