MTH 35: Vector Calculus Syllabus | Lecture notes Vector Analysis

Bronx Community College

of the City University of New York

Department of Mathematics and Computer Science

SYLLABUS: MTH 35: Vector Calculus (4 credits-4 hours)

Prerequisite: MTH 33: Calculus III

Main Text: Calculus, 6th Ed., by J. Stewart

Supplementary Text on reserve at the library: Vector Calculus, 5th Edition, by J.E.Marsden, A.J. Tromba

Learning Objectives: On successful completion of this course, students will be able to

•Compute the derivative matrix of multivariable and multivalued functions; state and apply the Chain rule to compute partial

derivatives.

•State and aplly the Implicit and Inverse Function theorems to determine if equations can be solved locally; find a derivative by

implicit methods.

•Find maxima and minima of functions subject to constraints using the Lagrange multipiers method.

•Compute double and triple integrals over general elementary regions in the plane and space respectively.

Compute the Jacobian of a transformation and use it to change variables in double and triple integrals; choose the appropriate

coordinate system (rectangular, polar, cylindrical, spherical) to compute double and triple integrals.

•Apply double and triple integrals to compute areas of surfaces, volumes of solids, averages, centers of mass, moments of inertia

and gravitational potentials.

•Compute line and surface integrals of functions and vector fields; use these integrals to compute work, flux, and mass integrals on

curves, surfaces and solids respectively.

•State and use the three vector forms of the Fundamental Theorem of Calculus in the multivariable setting: Green’s, Stokes’ and

Gauss’ (divergence) theorems.

•Determine whether a vector field is conservative; find and use potential functions to compute work integrals along curves.

•Use diﬀerential forms to state Green’s, Stokes’ and Gauss’ theorems under a single theorem.

•Apply Vector Analysis to study physical and geometrical problems described in the language of Partial Diﬀerential Equations.

This course addresses the following General Education Proficiencies: reasoning and analysis by a) understanding and applying

mathematical concepts, definitions and theorems in the formulation and solution of problems and b) by proving fundamental theorems;

mathematical and scientific methods to model, analyze and solve real-life problems in order to understand the physical, natural and

social worlds; communication skills by a) constructing, interpreting and applying symbolic and graphical representations of data and b)

by constructing and presenting (generally in writing, but, occasionally, orally) a rigorous mathematical argument; information literacy

by using technology appropriately to analyze and solve mathematical problems, for instance, by graphing vector fields, surfaces and,

in general, having students use technology to practice their skills at three-dimensional visualization, which is very important in this

course.

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Bronx Community College of the City University of New York Department of Mathematics and Computer Science

SYLLABUS: MTH 35: Vector Calculus (4 credits-4 hours) Prerequisite: MTH 33: Calculus III Main Text: Calculus, 6th Ed., by J. Stewart Supplementary Text on reserve at the library: Vector Calculus, 5th Edition, by J.E.Marsden, A.J. Tromba

Learning Objectives: On successful completion of this course, students will be able to

Compute the derivative matrix of multivariable and multivalued functions; state and apply the Chain rule to compute partial derivatives.
State and aplly the Implicit and Inverse Function theorems to determine if equations can be solved locally; find a derivative by implicit methods.
Find maxima and minima of functions subject to constraints using the Lagrange multipiers method.
Compute double and triple integrals over general elementary regions in the plane and space respectively. Compute the Jacobian of a transformation and use it to change variables in double and triple integrals; choose the appropriate coordinate system (rectangular, polar, cylindrical, spherical) to compute double and triple integrals.
Apply double and triple integrals to compute areas of surfaces, volumes of solids, averages, centers of mass, moments of inertia and gravitational potentials.
Compute line and surface integrals of functions and vector fields; use these integrals to compute work, flux, and mass integrals on curves, surfaces and solids respectively.
State and use the three vector forms of the Fundamental Theorem of Calculus in the multivariable setting: Green’s, Stokes’ and Gauss’ (divergence) theorems.
Determine whether a vector field is conservative; find and use potential functions to compute work integrals along curves.
Use differential forms to state Green’s, Stokes’ and Gauss’ theorems under a single theorem.
Apply Vector Analysis to study physical and geometrical problems described in the language of Partial Differential Equations.

This course addresses the following General Education Proficiencies: reasoning and analysis by a) understanding and applying mathematical concepts, definitions and theorems in the formulation and solution of problems and b) by proving fundamental theorems; mathematical and scientific methods to model, analyze and solve real-life problems in order to understand the physical, natural and social worlds; communication skills by a) constructing, interpreting and applying symbolic and graphical representations of data and b) by constructing and presenting (generally in writing, but, occasionally, orally) a rigorous mathematical argument; information literacy by using technology appropriately to analyze and solve mathematical problems, for instance, by graphing vector fields, surfaces and, in general, having students use technology to practice their skills at three-dimensional visualization, which is very important in this course.

Derivative of maps from Rn^ to Rm(mostly from the Supplementary Text) 1.5 n-Dimensional Euclidean Space p.86 1,2,4,7,8,10,11-15, 2.3 Definition of Differentiability for maps from Rn^ to Rm^ p.139 1-4,7,8,13, 3.4 & 15.8(from the Main Text) Lagrange Multipliers p.976 1-19 odd, 21-39 odd, 45 3.5 The Implicit Function Theorem p.253 1-3,5,7,9,10,

16 Multiple Integrals 16.1 Double Integrals over Rectangles p.994 1-17 odd 16.2 Iterated Integrals:Fubini’s Theorem p.1000 1-29 odd, 16.3 Double Integrals over General Regions p.1008 1-27 odd, 37-53 odd, 57 16.4 Double Integrals in Polar Coordinates p.1014 1-31 odd, 36, 16.5 Applications of Double integrals p.1024 1-19 odd 16.6 Triple Integrals p.1034 3-19 odd, 31 16.7 Triple Integrals in Cylinrdical Coordinates p.1040 1-25 odd, 27 16.8 Triple Integrals in Spherical Coordinates p.1046 1-31 odd, 35 16.9 Change of Variables in Multiple Integrals p.1056 1-19 odd,20, Review p.1057-p.

17 Vector Integral Calculus 17.1 Vector Fields p.1068 1-31 odd,35- 17.2 Line Integrals p.1079 1-21 odd,37,39,41, 17.3 The Fundamental Theorem for Line Integrals p.1089 1-23 odd,27,33, 17.4 Green’s Theorem p.1096 1-21 odd 17.5 Curl and Divergence p.1104 1-21 odd, 12, 25-31, 33— 17.6 Parametrized Surfaces and Their Areas p.1114 1,3,11-23 odd,33-45 odd 17.7 Surface Integrals of Scalar Functions and Vector Fields p.1127 1,5-27 odd,36,37-43 odd 17.8 Stokes’ Theorem p.1133 1-11a) odd,13,15, 17.9 The Divergence Theorem p.1139 1-13 odd,23,25- 8.5(from the Suplementary Text) Applications to Some Partial Differential Equations p.585 1,3,7- 8.6(from the Supplementary Text) Differential Forms p.603 1,3,4,5,6,8,9,11, Review p.1142-p.

MP (Spring 2010)

MTH 35: Vector Calculus Syllabus, Lecture notes of Vector Analysis

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