| CODE | MAT2512 | ||||||||
| TITLE | Vector Analysis 1 | ||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Double integrals; - Triple integrals; - Applications of multiple integrals; - Change of variables in multiple integrals; - Jacobian; - Grad, div and curl operators; - Space curves; - Serret-Frenet formulae; - Line integral and its applications; - Green’s theorem; - Conservative vector fields, scalar potential. Study-Unit Aims: • Meaning and application of multiple integrals including double and triple integration; • The Jacobian and its use in change of variables in multiple integrals; • Physical application of multiple integrals, such as the calculation of centre of mass; • Meaning and application of the gradient, divergence and curl operators; • The directional derivative and its application in physical problems; • Planar and space curves and their parameterization; • The geometry of curves, including curvature and torsion; • Derivation and application of the Serret-Frenet formulae; • Integration of a function along a curve and its applications; • Work done by a variable force acting on a particle moving along a curve; • Derivation and application of Green’s theorem for closed planar curves; • Conservative vector fields, scalar potential function and their applications. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit he student will be able to: • Solve problems involving double and triple integrals and apply change of variables when necessary; • Express the mass of an object and its centre of mass in terms of multiple integrals; • Apply the gradient, divergence and curl operators to scalar and vector fields and to construct other composite operators; • Obtain the parameterization of planar and space curves; • Understand the basic notion of the geometry of curves including curvature and torsion and the Serret-Frenet formulae; • Integrate a function along a curve in two or three dimensions; • Find the work done by a variable force acting on a particle moving along a trajectory; • Apply Green’s theorem to express line integrals in terms of double integrals on planar regions bounded by a finite number of boundaries; • Understand the meaning of conservative vector fields. 2. Skills By the end of the study-unit he student will be able to: • Visualize and draw a region in two or three dimensions involving a double or a triple integral and obtain the limits of integration from the region; • Decide when to apply change of variables and deduce the transformation equations representing the change of variables that will be most suited for the given problem; • Use basic operators, like the gradient, divergence and curl to construct composite operators like the Laplacian operator; • Obtain the parameterization of a curve from its explicit equation, and in particular find the arc length parameterization; • Calculate the curvature and torsion of a curve and determine whether a curve is planar or not; • Visualize a problem involving a line integral and express the line integral in terms of a single integral with respect to the parameter; • Determine when to use Green’s theorem to evaluate a line integral by expressing it in terms of a double integral; • Determine how to check whether a vector field is conservative and to evaluate line integrals involving conservative vector fields. 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: MAT1511 Leads to: MAT2513 |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
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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 2023/4. It may be subject to change in subsequent years. |
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