| CODE | MAT2513 | ||||||||
| TITLE | Vector Analysis 2 | ||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - The Laplacian and other operators; - Vector identities; - Introduction to surfaces; - Quadrics; - Surface integrals; - Applications of surface integrals; - Gauss’ theorem; - Stokes’ theorem; - Orthogonal curvilinear coordinates. Study-Unit Aims: • Further operators including the Laplacian and their use in vector equations; • The application of vector identities to simplify expressions and vector equations; • Basic geometry of a surface including its parameterization; • The first fundamental form of a surface and its applications; • General quadrics including some specific examples of commonly used quadrics; • Reduction of the equation of a quadric in standard form; • The construction and meaning of a surface integral; • Evaluation of a surface integral using the parametric equations or the explicit equation of a surface; • Applications of surface integration, as in the calculation of the surface area of a surface or the mass of a shell; • Derivation of Gauss’ and Stokes’ theorems and their applications for surfaces; • General orthogonal curvilinear coordinates and their properties; • Examples of orthogonal curvilinear coordinates such as the spherical polar coordinate system and the cylindrical polar coordinate system. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: • Use vector identities to evaluate or simplify vector expressions; • Know how to use the standard parameterizations of simple surfaces, such as the sphere, cylinder, cone, etc.; • Use the parameterization of a curve to obtain the parameterization of the corresponding surface of revolution; • Obtain the first fundamental form of a surface; • Know the standard equation and shape of simple quadrics; • Reduce the equation of a quadric in standard form; • Understand the meaning of a surface integral and its applications; • Use the explicit equation or the parametric equation of a surface to evaluate a surface integral; • Apply Gauss’ and Stokes’ theorem to problems involving surfaces; • Know the properties of general orthogonal curvilinear coordinates. 2. Skills By the end of the study-unit he student will be able to: • Obtain further identities from the standard vector identities; • Write down Laplace’s equation in rectangular Cartesian coordinates; • Obtain different parameterizations for the same surface and find the explicit equation of a surface from its parametric equation; • Find the tangents and normal to a surface; • Use the first fundamental form of a surface to find the surface area of a surface; • Find the length of a curve on a surface; • Apply the diagonalization theorem to obtain the canonical equation of a quadric; • Use surface integrals in physical problems such as to obtain the flux of a vector field across a given surface; • Use Gauss’ theorem to relate surface integrals and volume integrals and Stokes’ theorem to relate surface integrals and line integrals; • Obtain the components of a vector field in orthogonal curvilinear coordinates; • Write down the expressions for the gradient, divergence, curl and Laplacian operators in orthogonal curvilinear coordinates. Suggested Reading: - Finney R.L., Weir M.D. and Giordano F.R., Thomas Calculus, 10th Edition, Addison-Wesley Longman,New York, 2001. - Camilleri C.J., Vector Analysis, Malta University Press, 1994. - Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997. |
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| ADDITIONAL NOTES | Follows from: MAT2512 Leads to: MAT3221 |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Joseph Sultana |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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