Math 18.02 Concourse – Calendar of topics and HW assignments -- Fall 2018
The topics and assignments will change as the course proceeds!! Check back frequently. Last updated December 13, 2018 12:53 PM
|Seq.||Date||Topics||Assignments & References [Solutions]|
|L1||Wed, Sept 5||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 12)
Lecture #1 Notes (the sequence is all that matters)
|R1||Thurs, Sept 6||Parameterization of a line. Coordinate-free vector proofs.|
|L2||Mon, Sept 10||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 11||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 19)
Read sections 12.3 and 12.4 and the
|L3||Wed, Sept 12||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 13||Matrix multiplication and inverse matrices.|
|L4||Mon, Sept 17||
Matrix methods, continued. Parameterized curves in R2 and R3 (vector-valued functions); velocity vectors. [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 26)
|R4||Tues, Sept 18||
Problem set questions.
|L5||Wed, Sept 19||
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. Vector-valued functions and applications in physics. Parameterized surfaces vs. surfaces described by equations; cylinders, spheres, hyperboloids, ellipsoids, etc.
|R5||Thurs, Sept 20||Limits and continuity of functions of several variables. Introduction to partial derivatives.||
Read section 12.7 (Quadric Surfaces)
Problem Set #4 (due Fri, Oct 5)
|L6||Mon, Sept 24||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. Tangent plane to the graph of a function of two variables, linear approximation, differentials. [Lecture #6 Notes]|
|R6||Tues, Sept 25||
Exam topics: 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.
|L7||Wed, Sept 26||Rate of change of a function along a parameterized path (basic Chain Rule). Gradients; 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.|
|R7||Thurs, Sept 27||
Implicit differentiation - a new perspective. Supplement on the Chain Rule and Implicit Differentiation. Applications in economics. Higher order derivatives, equality of mixed partials; interpretation of 2nd derivatives with contour diagrams.
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 Fri, Oct 12)
|L8||Mon, Oct 1||
Directional derivatives. General Chain Rule.
|R8||Tues, Oct 2||
Exam #1 Solutions
|L9||Wed, Oct 3||Quadratic approximation, extrema of functions, stationary points and the 2nd Derivative test; local maxima, local minima, and saddle points. Unconstrained optimization.|
|R9||Thurs, Oct 4||Unconstrained optimization; Method of Least Squares and data fitting.|
|L10||Wed, Oct 10||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 18)
|R10||Thurs, Oct 11||Economics applications and the meaning of the Lagrange Multiplier. Partial derivatives with internal constraints (non-independent variables).|
|L11||Mon, Oct 15||Optimization with multiple constraints. Last words on optimization (bounded regions, multiple constraints). Partial derivatives with internal constraints (using differentials).||
Exam #2 will cover 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, the Method of Lagrange Multipliers, and (possibly) derivatives involving non-independent variables.
|R11||Tues, Oct 16||Questions and answers|
|L12||Wed, Oct 17||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.||
Problem Set #7 (due Thurs, Nov 1)
|R12||Thurs, Oct 18||
|L13||Mon, Oct 22||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.|
|R13||Tues, Oct 23||Exam #2 Solutions|
|L14||Wed, Oct 24||
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.
|R14||Thurs, Oct 25||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.||
Problem Set #8 (due Thurs, Nov 8)
|L15||Mon, Oct 29||Weighted averages and center of mass; other applications and examples. Cylindrical coordinates and spherical coordinates in triple integrals. Moment of inertia.|
|R15||Tues, Oct 30||
Problem set questions
|Here's a website that has a good java-based 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.]|
|L16||Wed, Oct 31||
Surface area. General change of variables in multiple integrals, Jacobian determinants. Parameterized surfaces - planes, spheres, cylinders, graphs. “Little (displacement) vectors” and “little patches” determined by varying parameters independently; scalar and vector elements of surface area.
15.2 (Line Integrals)
15.4 (Green's Theorem)
Problem Set #9 (due Tues, Nov 20)
|R16||Thurs, Nov 1|
|L17||Mon, Nov 5||
More on vector fields in R2 and R3, examples of vector fields from physics.
|R17||Tues, Nov 6|
|L18||Wed, Nov 7||
Integration along a parameterized path. Introduction to vector fields in R2 and R3, work done by a variable force along a parameterized path.
|R18||Thurs, Nov 8|
|R19||Tues, 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.
|L19||Wed, Nov 14||
Div, Grad, Curl; Green's Theorem; equivalents to a vector field being conservative.
|R20||Thurs, Nov 15||Exam #3 - Topics include: 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. Solutions|
|L20||Mon, Nov 19||
Examples of Green's Theorem; normal form of Green's Theorem.
14.8 (Surface Area)
Problem Set #10 (due Thurs, Nov 29)
|R21||Tues, Nov 20||
Integration on surfaces; surface area, average value, flux of a vector field through a surface.
|L21||Wed, Nov 21||
Examples of calculation of surface area and flux for graphs, cylinders, spheres; surface integrals for any parameterized surface. Supplement on surface integrals
|L22||Mon, Nov 26||
The Divergence Theorem and examples. Geometric (coordinate-free) definition of the divergence of a vector field.
15.7 (Stokes' Theorem)
Problem Set #11 (due Fri, Dec 7)
|R22||Tues, Nov 27|
|L23||Wed, Nov 28||Geometric (coordinate-free) definition of the divergence of a vector field; proof of the Divergence Theorem from the geometric definition of divergence.|
|R23||Thurs, Nov 29||
|L24||Mon, Dec 3||
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 4||Practice Questions for Exam #4 Solutions|
|L25||Wed, Dec 5||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 6||
Exam #4 Solutions
|L26||Mon, Dec 10||
Some topological matters; Maxwell's Equations; differential forms and other perspectives.
|R26||Tues, Dec 11||
Gradient, divergence, curl in other (orthogonal) coordinates
RW-Orthogonal Curvilinear Coordinates - Div, grad, curl, and Laplacian
|L27||Wed, Dec 12||Last details and questions|
|Thurs, Dec 20||FINAL EXAM, 9:00am to noon in 16-160|
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