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)

Lecture #2 Notes

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

Lecture #3 Note

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)

References:
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).

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)

Tangent plane to the graph of a function of two variables, linear approximation, differentials.

Lecture #6 Notes     Lecture #7 Notes

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

P-set questions; directional derivatives.

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.

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)

Lecture #7 Notes

Lecture Notes on Extrema of Functions of Several Variables

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.

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

Lecture #9 Notes

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)

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

Problem Set questions and intoduction to triple integrals and their applications; weighted averages and center of mass; other applications and examples.

Lecture #10 Notes

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)

Problem set questions and intoduction to cylindrical coordinates and spherical coordinates in triple integrals. Moment of inertia.

Cylindrical coordinates and spherical coordinates in triple integrals. Moment of inertia.

Lecture #10 Notes

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)

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

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

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)

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

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

Lecture #13 Notes

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

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)

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.

Examples, definitions of gradient and divergence in other (orthogonal) coordinate systems.

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

Some topological matters; Maxwell's Equations; differential forms and other perspectives.

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