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Module MA2321: Analysis in several real variables
- Credit weighting (ECTS)
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5 credits
- Semester/term taught
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Michaelmas term 2016-17
- Contact Hours
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11 weeks, 3 lectures including tutorials per week
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- Lecturer
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Prof. David Wilkins
- Learning Outcomes
- On successful completion of this module, students will be able to:
- justify with logical argument basic results concerning the
topology of Euclidean spaces, convergence of sequences
in Euclidean spaces, limits of vector-valued functions
defined over subsets of Euclidean spaces and the continuity
of such functions;
- specify accurately the concepts of differentiability
and (total) derivative for functions of several
real variable;
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- justify with logical argument basic properties of
differentiable functions of several real variables
including the Product Rule, the Chain Rule, and the
result that a function of several real variables is
differentiable if its first order partial derivatives
are continuous;
- determine whether or not specified functions of
several real variables satisfy differentiability
requirements.
- Module Content
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- Review of real analysis in one real variable: the real number system; the least upper bound axiom; convergence of monotonic sequences; the Bolzano-Weierstrass Theorem in one variable; the Extreme Value Theorem; differentiation; Rolle's Theorem; the Mean Value Theorem; Taylor's Theorem; the Riemann integral.
- Analysis in several real variables: convergence of sequences of points in Euclidean spaces; continuity of vector-valued functions of several real variables; the Bolzano-Weierstrass Theorem for sequences of points in Euclidean spaces; the Extreme Value Theorem for functions of several real variables.
- Differentiablity for functions of several variables: partial and total derivatives; the Chain Rule for functions of several real variables; properties of second order partial derivatives.
- Module Prerequisite
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- Recommended Reading
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- Principles of Real Analysis, W. Rudin. McGraw-Hill, 1976;
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- Assessment Detail
- This module will be examined in a 2-hour examination
in Trinity term. Continuous assessment will
contribute 10% to the final grade for the module at the
annual examination session, with the examination
counting for the remaining 90%. The supplemental
examination paper, if required, will determine
100% of the supplemental module mark.