MA4006: Engineering Mathematics V Vector Calculus and Partial ...

MA4006: Engineering Mathematics 5 Vector Calculus and Partial Differential Equations Course outline and schedule Spring Semester, 2014/15 • Lecturer: Dr. Sarah Mitchell ([email protected], office B3042). • Office hours: Mondays 10-11am, Wednesdays 10-11 (or by appointment, email to arrange). • Lectures: Mondays 1-2pm (room HSG037), Tuesdays 5-6pm (room P1033) and Wednesdays 1-2pm (room FB028). • Tutorials (starting week 2): Group 3A, Mondays 9-10am (room KBG13); Group 3B, Wednesdays 3-4pm (room SG15); Group 3C, Thursdays 12-1pm (room KBG13). • Webpage: http://www.staff.ul.ie/mitchells/MA4006.html. Important notices, lecture notes and exercise sheets (including solutions) will be posted here. • Midterms: There will be two 50 minute midterms, each worth 10%, on Monday 2nd March (week 6) and Monday 6th April (week 10). • Lecture notes: Printed notes will be available on the course webpage. Extra examples will be given in lectures that are not in these notes. • Exercise sheets: These will be put on the webpage every other week (end of week 1, week 3 etc.) and these should be attempted before the tutorials the following two weeks (i.e. weeks 2 and 3, weeks 4 and 5 etc.) • Aims/objectives: – Vector Calculus: scalar and vector fields, contour maps, directional derivative and gradient vector of a scalar field, divergence and curl of a vector field (line, surface and volume integrals), Gauss’, Green’s and Stokes’ Theorems. – PDEs: modelling and derivation of some linear PDEs (e.g. wave, heat and Laplace equations), solution by separation of variables, Laplace and Fourier transforms, numerical methods using finite difference schemes. • References: – (Primary text): Advanced Engineering Mathematics, Kreyszig; – Vector Analysis (Schaum series), Murray and Speigel; – Modern Engineering Mathematics, James; – Elementary Differential Equations and Boundary Value Problems, Boyce, Di Prima; – Elementary Partial Differential Equations, Gribben; – Solution of Partial Differential Equations, Duffy; – Advanced Engineering Mathematics, Stroud; – Integral transforms in Mathematical Physics, Tranter.

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Pre-requisite vector calculus • Scalar: an entity or property of a physical system which can be represented by a single real number, e.g. temperature of a body, mass of a body. • Vector: an entity with both a magnitude and a direction (and requiring three real numbers to fully specify it), e.g. the velocity of a car. • Scalar (dot, inner) product of two vectors a, b: a · b = |a||b| cos θ, −→ −−→ where θ is the angle between the the vectors OA and OB. Alternatively a · b = a1 b1 + a2 b2 + a3 b3 , where the vectors in cartesian co-ordinates are a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ), or a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k. • Cross (vector) product of two vectors a, b: a × b = |a||b| sin θ k, −→ −−→ where 0 ≤ θ ≤ π is the angle between the vectors OA and OB and k is a unit vector perpendicular to both a and b such that a, b, k form a right-handed triple. Alternatively   i j k a × b = det  a1 a2 a3  = (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). b1 b2 b3 • Triple scalar product  a1 a2 a3 a1 a2 a3 a · (b × c) = (a · b) × c = det  b1 b2 b3  = b1 b2 b3 . c1 c2 c3 c1 c2 c3 

• Determinants: 2 × 2: 3 × 3:



a1 a2 = a1 b2 − a2 b1 b1 b2 a1 a2 a3 b1 b2 b1 b3 b2 b3 b1 b2 b3 = a1 − a2 c1 c3 + a3 c1 c2 c c 2 3 c1 c2 c3 = a1 (b2 c3 − c2 b3 ) − a2 (b1 c3 − c1 b3 ) + a3 (b1 c2 − c1 b2 ).

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