Math 18.02 Concourse – Calendar of topics and HW assignments  Fall 2019
The topics and assignments will change as the course proceeds!! Check back frequently. Last updated
December 20, 2019 1:49 PM
Seq.  Date  Topics  Assignments & References [Solutions] 
L1  Wed, Sept 4  Course overview. Coordinates in R^{2} and R^{3}. Vectors in R^{2} and R^{3} and vector algebra; magnitude of a vector, unit vectors, difference vector, distance. Vector equation of a line and parametric equations of a line in R^{2} and R^{3}.  Read sections 12.1 and 12.2 in the Edwards & Penney text. Problem Set #1 (due Wed, Sept 11) References: Notes on dot products and cross products Lecture #1 Notes (the sequence is all that matters) 
R1  Thurs, Sept 5  Parameterization of a line. Coordinatefree vector proofs.  
L2  Mon, Sept 9  Dot product of two vectors; angle between vectors; Law of Cosines; orthogonality; scalar and vector projections using the dot product. Equations for lines and planes. Cross product of two vectors in R^{3}.  
R2  Tues, Sept 10  Planes and linear equations; cross product of two vectors in R^{3}, continued. Applications to areas, volumes; triple scalar product.  Problem Set #2 (due Wed, Sept 18) Read sections 12.3 and 12.4 and the D. Determinants (4 pages) Supplement on Solving Systems of Linear Equations via Row Reduction 
L3  Wed, Sept 11  Algebra and geometry of lines and planes. Matrix methods and systems of linear equations, row reduction and parameterization of lines and planes. [Lecture #3 Notes]  
R3  Thurs, Sept 12  Matrix multiplication and inverse matrices.  
L4  Mon, Sept 16  Parameterized curves in R^{2} and R^{3} (vectorvalued functions); velocity vectors; unit tangent vector, acceleration vector; arclength, acceleration vector, curvature. Calculus of vectorvalued functions.. [Java applet that may help you to understand parameterized curves: Wheel (choose the "Trace" option) and click on the double arrows to activate.] 
Problem Set #3 (due Wed, Sept 25) References: 
R4  Tues, Sept 17  Problem set questions.  
L5  Wed, Sept 18  Vectorvalued functions and applications in physics. Parameterized surfaces vs. surfaces described by equations; cylinders, spheres, hyperboloids, ellipsoids, etc. 
Read section 12.7 (Quadric Surfaces) Problem Set #4 (due Thurs, Oct 3) 
R5  Thurs, Sept 19  Limits and continuity of functions of several variables. Introduction to partial derivatives.  
L6  Mon, Sept 23  Functions of several variables and their level sets. Graphs and contours (level sets) of a function. Rate of change of a function of several variables  partial derivatives. [Lecture #6 Notes]  
R6  Tues, Sept 24  Tangent plane to the graph of a function of two variables, linear approximation, differentials. 

L7  Wed, Sept 25  Rate of change of a function along a parameterized path (basic Chain Rule); gradients. Notes on the gradients, Chain Rule, implicit 
Practice Questions for Exam #1 Solutions (use same username/password as solutions) 
R7  Thurs, Sept 26  Pset questions; directional derivatives. 

L8  Mon, Sept 30  Exam #1  Vectors and vector algebra, dot product, cross product, applications to working with lines and planes, areas, volumes, angles, etc.; parameterized curves, velocity vectors, speed, arclength, unit tangent and normal vectors; partial derivatives, linear approximation, differentials, and directional derivative. You should also be familiar with methods for solving systems of linear equations (such as when finding the intersection of lines or planes) and related matrix ideas. 

R8  Tues, Oct 1  Using the gradient to find a normal vector to a curve or surface and equations for tangent lines to curves and tangent planes to surfaces. General Chain Rule; Implicit differentiation  a new perspective. Supplement on the Chain Rule and Implicit Differentiation. Applications in economics  marginal rate of substitution (MRS). 
Read sections 13.7 (Chain Rule and Implicit Differentiation), 13.8 (Directional Derivatives and the Gradient); the Supplement on the Chain Rule and Implicit Differentiation; and Supplementary Notes TA (Tangent Approximation), 13.10 (Critical Points of Functions of Two Variables and the Second Derivative Test); and Supplementary Notes LS (Least Squares Interpolation). Problem Set #5 (due Thurs, Oct 10) 
L9  Wed, Oct 2  Higher order derivatives, equality of mixed partials; interpretation of 2nd derivatives with contour diagrams. Quadratic approximation, extrema of functions, stationary points and the 2nd Derivative test; local maxima, local minima, and saddle points.  
R9  Thurs, Oct 3  Unconstrained optimization. Method of Least Squares and data fitting.  
L10  Mon, Oct 7  Constrained optimization and the Method of Lagrange Multipliers. Extrema in a bounded region; [Java tool for Lagrange Multipliers]  Read sections 13.5 (Multivariable Optimization Problems), 13.9 (Lagrange Multipliers and Constrained Optimization). Problem Set #6 (due Thurs, Oct 17) 
R10  Tues, Oct 8  Economics applications and the meaning of the Lagrange Multiplier. Partial derivatives with internal constraints (nonindependent variables).  
L11  Wed, Oct 9  Optimization with multiple constraints. Last words on optimization (bounded regions, multiple constraints). 

R11  Thurs, Oct 10  Questions and answers. Partial derivatives with internal constraints (using differentials).  Read sections: Problem Set #7 (due Tues, Oct 29) 
L12  Wed, Oct 16  Introduction to integration of a function f (x, y) over a region in R^{2}; applications to volume, mass, population, average value of a function; centroid (geometric center) of a region.  
R12  Thurs, Oct 17  Applications of double integrals calculated by iterated single integrals (successive slicing) in Cartesian coordinates and polar coordinates. Weighted averages and center of mass; other applications and examples.  
L13  Mon, Oct 21  Calculation of integrals over regions R^{2} via iterated single integrals (successive slicing) in Cartesian coordinates. The Fubini Theorem; interchanging the order of integration in an iterated double integral. Calculation of integrals over regions R^{2} using polar coordinates.  
R13  Tues, Oct 22  
L14  Wed, Oct 23  Change of variables in double integrals. Introduction to integration of a function f (x, y, z) over a region in R^{3}  triple integrals to calculate mass, volume, average value, centroid, and more in R^{3} using Cartesian coordinates. 
Practice Exam #2 Practice Exam #2 solutions 
R14  Thurs, Oct 24  Problem Set questions and intoduction to triple integrals and their applications; weighted averages and center of mass; other applications and examples. 

L15  Mon, Oct 28  Exam #2  Partial derivatives, linear approximation, differentials, gradient vector, normal vectors to curves and surfaces, directional derivative, the Chain Rule, implicit differentiation, stationary points, second derivative test, optimization, Method of Lagrange Multipliers. Double integrals in Cartesian and polar coordinates and applications; Fubini Theorem and interchanging order of integration.  Read sections: Problem Set #8 (due Wed, Nov 13) 
R15  Tues, Oct 29  Problem set questions and intoduction to cylindrical coordinates and spherical coordinates in triple integrals. Moment of inertia. 

L16  Wed, Oct 30  Cylindrical coordinates and spherical coordinates in triple integrals. Moment of inertia. 

R16  Thurs, Oct 31  General change of variables in multiple integrals, Jacobian determinants.  
L17  Mon, Nov 4  Surface area. Parameterized surfaces  planes, spheres, cylinders, graphs. “Little (displacement) vectors” and “little patches” determined by varying parameters independently; scalar and vector elements of surface area. 
Read sections: 15.2 (Line Integrals) 15.3 (Fundamental Theorem and Independence of Path) 15.4 (Green's Theorem) Problem Set #9 (due Wed, Nov 20) 
R17  Tues, Nov 5  Here's a website that has a good javabased tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.]  
L18  Wed, Nov 6  Integration along a parameterized path. Introduction to vector fields in R^{2} and R^{3}, work done by a variable force along a parameterized path, examples of vector fields from physics. 

R18  Thurs, Nov 7  
R19  Tues, Nov 12 


L19  Wed, Nov 13  Calculation of line integrals. Independence of path and conservative (gradient) vector fields. Fundamental Theorem of Line Integrals and potential functions. Algebraic definitions of the divergence and curl of a vector field. 
Supplement on conservative vector fields and the Fundamental Theorem of Line Integrals 
R20  Thurs, Nov 14  Div, Grad, Curl; Green's Theorem; equivalents to a vector field being conservative.  
L20  Mon, Nov 18  Geometric, coordinatefree proof of Green's Theorem; normal form of Green's Theorem; circulation and 2Dflux. 
Read sections: 14.8 (Surface Area) Problem Set #10 (due Wed, Nov 27) 
R21  Tues, Nov 19  Examples of Green's Theorem and normal form of Green's Theorem; pset questions. 

L21  Wed, Nov 20  Integration on surfaces; surface area, average value, flux of a vector field through a surface. 

R22  Thurs, Nov 21  Examples of calculation of surface area and flux for graphs, cylinders, spheres; surface integrals for any parameterized surface. Supplement on surface integrals 
Exam #3 Practice Problems Solutions 
L22  Mon, Nov 25  Geometric (coordinatefree) definition of the divergence of a vector field; proof of the Divergence Theorem from the geometric definition of divergence. 
Read sections: 15.7 (Stokes' Theorem) Problem Set #11 (due Fri, Dec 6) 
R23  Tues, Nov 26  Exam #3  Integration over two and threedimensional regions; double and triple integrals in Cartesian, cylindrical, and spherical coordinates; Fubini Theorem and interchanging order of integration; applications of integration – areas, volumes, mass, averaging, weighted averages, centroids and center of mass, moment of inertia, general change of variables for double and triple integrals, Jacobian determinants; Integration of functions over curves and surfaces (line and surface integrals) with applications to mass, averaging, centroids, flux, etc.; flux integrals in 2 and 3dimensions; conservative vector fields and potential functions; Fundamental Theorem of Line Integrals, Green’s Theorem. 

L23  Wed, Nov 27  Examples, definitions of gradient and divergence in other (orthogonal) coordinate systems. 

L24  Mon, Dec 2  Stokes' Theorem and examples. Proof of Green's Theorem as a special case of Stokes' Theorem; proof that curl F = 0 implies F is conservative (given appropriate conditions). 

R24  Tues, Dec 3  
L25  Wed, Dec 4  Geometric (coordinatefree) definition of the curl of a vector field. Proof of Stokes' Theorem from the geometric definition of curl. Examples.  
R25  Thurs, Dec 5  Some topological matters; Maxwell's Equations; differential forms and other perspectives. 

L26  Mon, Dec 9  Gradient, divergence, curl in other (orthogonal) coordinates RWOrthogonal Curvilinear Coordinates  Div, grad, curl, and Laplacian 
Practice Exam A (from mainstream 18.02) Solutions Practice Exam B (from mainstream 18.02) Solutions 
R26  Tues, Dec 10 


L27  Wed, Dec 11  Last details and questions  
Tues, Dec 17  FINAL EXAM, 9:00am to noon in 16160 