Singapore University of Social Sciences

Analysis in Euclidean and Metric Spaces (MTH402)

Applications Open: To be confirmed

Applications Close: To be confirmed

Next Available Intake: To be confirmed

Course Types: To be confirmed

Language: English

Duration: 6 months

Fees: To be confirmed

Area of Interest: Science & Technology

Schemes: To be confirmed

Funding: To be confirmed


Synopsis

The techniques developed in MTH401 are exploited in spaces of continuous functions and applications are made to differential and integral equations. Differentiable aspects of vector-valued functions are then discussed.

Level: 4
Credit Units: 5
Presentation Pattern: Every July

Topics

  • The space of continuous functions.
  • Banach spaces. Equicontinuity and pointwise compactness.
  • Arzela-Ascoli Theorem.
  • Lipschitz condition.
  • Contraction Mapping Principle and applications to differential and integral equations.
  • Bernstein polynomials.
  • Stone-Weierstrass Theorem.
  • Differentiability and the derivative of vector-valued functions.
  • Jacobian matrix, gradient vectors.
  • Differentiable curves and tangent vectors.
  • Conditions for differentiability.
  • Directional derivatives. Chain rule. Product rule.

Learning Outcome

  • solve calculus problems between Euclidean spaces;
  • examine the type of local extrema of functions;
  • test the validity of various theorems in analysis;
  • verify that conditions in analysis are satisfied in order to apply major theorems;
  • construct matrix representations of the derivative of maps between Euclidean spaces;
  • construct from definition various types of derivatives of specific maps between Euclidean spaces;
  • construct proofs of normed, metric, completeness and topological structures, and various properties of given subsets of the space of continuous maps;
  • create proofs of various properties, identities and relations for continuous, contraction or differentiable mappings;
  • design iteration schemes that use contraction mappings to solve problems in differential or integral equations.
  • solve a number of problem sets within strict deadlines;
  • construct a range of mathematical techniques to solve a variety of formal problems;
  • formulate solutions to problems individually and/or as part of a team.
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