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 R2 and R3. Vectors in R2 and R3 and vector algebra; magnitude of a vector, unit vectors, difference vector, distance. Vector equation of a line and parametric equations of a line in R2 and R3.

Read sections 12.1 and 12.2 in the Edwards & Penney text.

Problem Set #1 (due Wed, Sept 11)


Notes on dot products and cross products

Lecture #1 Notes (the sequence is all that matters)

Lecture #2 Notes

R1 Thurs, Sept 5 Parameterization of a line. Coordinate-free 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 R3.
R2 Tues, Sept 10 Planes and linear equations; cross product of two vectors in R3, continued. Applications to areas, volumes; triple scalar product.

Problem Set #2 (due Wed, Sept 18)

Read sections 12.3 and 12.4 and the
Matrices and Linear Algebra Notes.

D. Determinants (4 pages)
Exercises for Vectors and Matrices (1A to 1K, 12 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.

Lecture #3 Note

L4 Mon, Sept 16

Parameterized curves in R2 and R3 (vector-valued functions); velocity vectors; unit tangent vector, acceleration vector; arclength, acceleration vector, curvature. Calculus of vector-valued functions.. [Java applet that may help you to understand parameterized curves: Wheel (choose the "Trace" option) and click on the double arrows to activate.]

[Lecture #4 Notes]

Problem Set #3 (due Wed, Sept 25)

Parameterized Curves (3 pages)
Read sections 10.4 (Parametric Curves), 12.5 (Curves and Motions in Space), 12.6 (Curvature and Acceleration), and appropriate sections from the Supplementary Notes (1F, 1G, 1H, 1J).

R4 Tues, Sept 17 Problem set questions.
L5 Wed, Sept 18

Vector-valued functions and applications in physics. Parameterized surfaces vs. surfaces described by equations; cylinders, spheres, hyperboloids, ellipsoids, etc.

Lecture #5 Notes

Mathlet (Java applet) for Curves and Surfaces

Read section 12.7 (Quadric Surfaces)
Read sections 13.1-13.2 (Functions of Several Variables);
13.3 (Limits and Continuity);
13.4 (Partial Derivatives);
13.6 (Increments and Linear Approximation);
Supplementary Notes TA (Tangent Approximation).

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.

Lecture #6 Notes     Lecture #7 Notes

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
differentiation, and higher order derivatives

Practice Questions for Exam #1     Solutions
(use same username/password as solutions)
R7 Thurs, Sept 26

P-set 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.

Lecture #7 Notes

Lecture Notes on Extrema of Functions of Several Variables

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]

Lecture #8 Notes

Read sections 13.5 (Multivariable Optimization Problems), 13.9 (Lagrange Multipliers and Constrained Optimization).

Problem Set #6 (due Thurs, Oct 17)
You may use any method unless a particular method is specified.

R10 Tues, Oct 8 Economics applications and the meaning of the Lagrange Multiplier. Partial derivatives with internal constraints (non-independent variables).
L11 Wed, Oct 9

Optimization with multiple constraints. Last words on optimization (bounded regions, multiple constraints).

Lecture #9 Notes

R11 Thurs, Oct 10 Questions and answers. Partial derivatives with internal constraints (using differentials).

Read sections:
SN-N. Non-independent Variables (8 pages)
14.1 (Double Integrals)
14.2 (Double Integrals Over More General Regions)
14.3 (Area and Volume by Double Integration)
14.4 (Double Integrals in Polar Coordinates)
14.5 (Applications of Double Integrals)
SN-I. Limits in Iterated Integrals (4 pages)

Problem Set #7 (due Tues, Oct 29)

L12 Wed, Oct 16 Introduction to integration of a function f (x, y) over a region in R2; 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 R2 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 R2 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 R3 - triple integrals to calculate mass, volume, average value, centroid, and more in R3 using Cartesian coordinates.

Lecture #10 Notes


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.

Lecture #10 Notes

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:
14.6 (Triple Integrals)
14.7 (Integration in Cylindrical and Spherical Coordinates)
14.8 (Surface Area)
14.9 (Change of Variables in Multiple Integrals)
SN-I. Limits in Iterated Integrals
SN-CV. Changing Variables in Multiple Integrals
SN-G. Gravitational Attraction

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.

Lecture #10 Notes

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.

Lecture #11 Notes

Read sections:
14.8 (Surface Area)
15.1 (Vector Fields, Divergence and Curl)
Notes V1 (Plane Vector Fields)
Notes V8 (Vector Fields in Space)

15.2 (Line Integrals)
Notes V11 (Line Integrals in Space)

15.3 (Fundamental Theorem and Independence of Path)
Notes V2 (Gradient Fields and Exact Differentials)
Notes V12 (Gradient Fields in Space)

15.4 (Green's Theorem)

Problem Set #9 (due Wed, Nov 20)

R17 Tues, Nov 5 Here's a website that has a good java-based tool for showing vector fields and flows: 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 R2 and R3, work done by a variable force along a parameterized path, examples of vector fields from physics.

Lecture #11 Notes

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, coordinate-free proof of Green's Theorem; normal form of Green's Theorem; circulation and 2D-flux.

Lecture #12 Notes

Read sections:
Notes V3 (Two-dimensional Flux)
Notes V4 (Green's Theorem in Normal Form)

14.8 (Surface Area)
Notes V7 (Laplace's Equation and Harmonic Functions)
15.5 (Surface Integrals)
Notes V9 (Surface Integrals)
RW-Supplement on surface integrals
15.6 (The Divergence Theorem)
Notes V10 (The Divergence Theorem)

Problem Set #10 (due Wed, Nov 27)

R21 Tues, Nov 19

Examples of Green's Theorem and normal form of Green's Theorem; p-set questions.

L21 Wed, Nov 20

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

Lecture #13 Notes

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

Lecture #14 Notes

Exam #3 Practice Problems     Solutions

L22 Mon, Nov 25

Geometric (coordinate-free) definition of the divergence of a vector field; proof of the Divergence Theorem from the geometric definition of divergence.

Lecture #14 Notes

Read sections:
15.6 (The Divergence Theorem)
Notes V10 (The Divergence Theorem)
RW-Orthogonal Curvilinear Coordinates - Div, grad, curl, and Laplacian

15.7 (Stokes' Theorem)
Notes V13 (Stokes' Theorem)
Notes V14 (Some Topological Questions)
Notes V15 (Relations to Physics)

Problem Set #11 (due Fri, Dec 6)

R23 Tues, Nov 26

Exam #3 - Integration over two- and three-dimensional 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 3-dimensions; 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).

Lecture #14 Notes

R24 Tues, Dec 3  
L25 Wed, Dec 4 Geometric (coordinate-free) 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

RW-Orthogonal Curvilinear Coordinates - Div, grad, curl, and Laplacian

Practice Exam A (from mainstream 18.02)     Solutions

Practice Exam B (from mainstream 18.02)     Solutions

Mega-List of Math 18.02 techniques

Math 18.02 Useful Facts

R26 Tues, Dec 10


L27 Wed, Dec 11 Last details and questions
  Tues, Dec 17 FINAL EXAM, 9:00am to noon in 16-160

Return to main Math 18.02 Concourse page