# MAT4223

## Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
7.1
• Understand the difference between pointwise convergence and uniform convergence
• Understand how the continuity, diferrentiablity and integrability behave under uniform convergence
Week 1
7.2
• Learn the term-by-term differentiation (resp. integration) theorem
• Weierstrass M-test
Week 2
7.3 and 7.4
• Determine the set of convergence for a power series
• Apply the results of the previous sections to power series.
• Apply the results to the elementary functions
Week 2
7.5
• Learn to construct a continuous function everywhere which is not differentiable anywhere

Week 3
8.1
• Norms in the Euclidean space, comparison
• Cauchy-Schwartz inequality, triangle inequality

Week 3
8.2
• The matrix of a linear transformation
• The norm of a linear transformation
• Write equations of hyperplanes in the Euclidean space

Week 3
8.3 and 8.4
• Learn basic notions of topology in the Euclidean space

Week 4
9.1
• Convergence theorems
• Bolzano-Weierstrass Theorem

Week 4
9.2
• Understand the structure of compact sets
• Applications

Week 5
9.3
• Limit theorems, sequential criterion
• Iterated limits
• Commutation of iterated limits

Week 5
9.4
• Characterizations of continuity
• Properties of continuous functions
• Extreme value theorem

Week 6
11.1
• Commutation of partial derivatives
• Commutation of the limit with the the integral
• Differentiating under the integral sign

Week 7
11.2
• Necessary and sufficient conditions for differentiation
• The total derivative of a differentiable function
• Determine whether a function if differentiable or not

Week 7
11.3 and 11.4
• Basic rules for differentiation and applications
• Tangent hyperplanes to surfaces
• Chain rule

Week 8
11.5
• Mean-value theorems for vector-valued functions
• Lipschitz functions

Week 8
11.5 and 11.6
• Taylor's Formula in several variables, applications

Weeks 8 and 9
11.6
• The inverse function theorem and the implicit function theorem for vector-valued functions
• Applications

Week 10
9.6
• Define an almost everywhere continuous function
• The Riemann integrable functions are those almost everywhere continuous

Weeks 11 and 12
12.1
• Jordan regions: volume and properties

Weeks 13 and  14
12.2
• Definition of a Riemann integrable function, characterization, properties
• Any continuous function on a closed Jordan region is integrable
• Mean-valued theorems for multiple integrals
Week 15
12.3
• Fubini's theorem and applications