**Module leader**: Felix Schulze **Lectures**: Monday 3-4pm, MS.02; Tuesday 10-11am, MS.02; Friday 9-10am, MS.02 **Tutorials**: Weeks 2-10, Monday 13-14, B3.03; Monday 16-17, L4; Thursday 13-14, MS.03 **Due Dates**: 4 assignments, due at noon on the Monday of weeks 4, 6, 8 and 10. **Workspace**: [[MA263 Multivariable Analysis Workspace.canvas|MA263 Multivariable Analysis Workspace]]. **Course Summary**: ... --- # 2. Functions on Euclidean Space ### 2.1 Norm and Inner Product | Examples | Theorems | Definitions/ Lemmas | | -------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------- | | | 2.1.1. [[Euclidean Norm]] and [[Inner Product is At Most Product of Vector Norms (Cauchy-Schwartz Inequality)]].<br><br>2.1.2 [[Dot Product is an Inner Product on Real n-Space]] and [[Euclidean Norm Satisfies Polarization Identity]]. | | ### 2.2 The space of Linear maps and matrices ### 2.3 Subsets of Euclidean Space | Examples | Theorems | Definitions/ Lemmas | | -------- | -------------------------------------------------------------------------------------------------- | ------------------- | | | 2.3.1 [[Subset of Euclidean space is sequentially compact iff closed and bounded]] and ... | | ### 2.4 Functions and Continuity | Examples | Theorems | Definitions/ Lemmas | | ----------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------- | | | 2.4.2 [[Characterization of Continuity Via Open Sets]].<br><br>2.4.3. [[Continuous Functions Preserve Sequentially Compact Subsets of Euclidean Space]].<br><br>2.4.4 [[Composition of Continuous Functions in Continuous]]. | 2.4.1 [[Continuous maps]]. | | | | | | [[Linear Maps are Lipschitz Continuous]]. | | | # 3. Differentiation See [[Fréchet Differentiation]]. # 4. The Inverse Function Theorem See [[Inverse function theorem]]. # 5. The Implicit Function Theorem See [[Implicit function theorem]]. # 6. Second Order Derivatives # 7. Integration 7.1/2. [[Riemann integration]]. 7.3. [[Fubini's theorem]] 7.4. [[Existence of smooth partition of unity of subset of Euclidean space]]., See [[Change of variables formula]]. # 8. The Divergence Theorem See [[Divergence theorem]].