Vector Calculus - Study Guide - Final Exam

Nature of the exam

The final assessment is cumulative and tests whether you have learned the concepts and derivations in the packets, and the techniques used in the exercises and problems. There will not be an intentional emphasis on material from Packets 11-15, rather all packets will be equally represented (or equally likely to be represented).

The format will consist of a single part with a simple sequence of problems on topics that progress through the packets, in order. For each problem, the first 1-3 sub-questions will focus on memory of definitions, formulas, or derivations, while the last 1-2 sub-questions will ask for a calculation or solution to a small problem. Many such problems will be drawn from the Exercises and Problems of the course, but that is not guaranteed for all of them.

One of the problems is guaranteed to ask for a complete derivation of either (i) curl as infinitesimal circulation, or (ii) divergence as infinitesimal expansion, both using linear approximation on a small circle. It is strongly recommended that you create for yourself a summary outline of the derivation, similar to the one for Kepler’s Laws in the packets. This is how you learn a derivation! Your summary should be a short sequence of “heading” phrases (~3-5), each phrase having the keywords of a step of the solution. During the exam, write this down first, and then unpack the details from your headings.

Study tips

I recommend this study method: Create your own set of complete notes

• Very condensed, summary level, comprehensive coverage, as if for a cheat-sheet.
• Write out all definitions and formulas, and illustrate them with example calculations.
• Include boxes containing longer derivations, examples, and problem solutions, placed under the theory elements that they rely on.
• Spend time studying these notes to memorize formulas and learn everything! You should spend at least as much time studying the notes as you did creating them.
• While studying it is probably helpful to write a second copy of the notes in more compressed form (only 30% the length of your first notes); compressing the notes is a helpful activity, and the compressed version is easier to memorize.
• To check your memory, give yourself short prompts, and try to unpack full details on paper or aloud, then check against your notes.

Meet with your Group members to study for the final! Review each other’s notes and quiz each other on concepts and problems.

Theory List

Packet 02

• Parametric equation of a line
• Parametric curves vs. graphs of functions
• Parametric curves and algebraic equations satisfied by their image
• Formula to compute the traditional derivative
• Formula to compute the traditional second derivative
• Derivation of both formulas formula for first derivative
• Parametric tangent line to parametric curve
• Distance traveled, speed, length of a curve
• Planar curvature definition
• Planar curvature formula:
• Polar coordinates, plotting
• Area under polar graph:

Packet 03

• Know these vector space rules: Commutativity (addition), Associativity (addition), Distributivity (scalars)
• Dot product: definition, algebraic rules, angle formula with derivation
• Norm definition, and determination of dot product from norm
• Projection : meaning, formula, meaning of formula
• Division of vector into parallel and perpendicular parts
• Cauchy-Schwarz identity, Triangle identity (no proofs, just identities and ability to verify in examples)
• Right- and left-handedness for ordered triples of orthogonal unit vectors
• Cross product: definition by components, basic algebraic properties, angle formula
• Relation of cross product to right-handedness
• Dot-cross triple product: geometric meaning
• Cross-cross triple product: “bac-cab” rule for Lagrange identity
• Planes: vector form and scalar form, distance plane to point

Packet 04

• Surfaces: classify an equation (without cross-terms)
  • complete the square
  • put into standard form
  • decide what kind of surface it is
• Write down and describe the trace of a surface as a conic section (i.e. one variable being a fixed constant)

Packet 05

• Componentwise differentiation of parametric curves
• Tangent vector interpretation and application to find tangent lines
• Derivative rules, including proof of product rules
• Trick fact: constant norm means perpendicular to its derivative (with proof)
• Definitions of , , , , , , ,
• Decomposition of acceleration; meaning of and and calculation of them
• Efficient curvature formula: derivation and application
• Efficient normal vector formula and/or derivation
• Arc length parametrization: definition and how to compute (simple applications)

Packet 06

• Vector integration: definite and indefinite
• Solving basic differential equations with integration, initial conditions
• Kepler’s Law of Ellipses: be able to reproduce all or any part of this derivation from memory
• Kepler’s other laws
• Scalar line integrals (both notations!)
• Creation of parametric curves e.g. of lines (between points, or point-and-velocity)

Packet 08

• Iterated integral calculations, including for non-rectangular regions (variable bounds)
• Fubini Theorem: name, statement for rectangular regions, ability to verify the result for specific integrands
• Volume under a surface over some region in the -plane
• Changing order with variable bounds: sketch region and write new bound functions
• Integration in polar or spherical coordinates: and
• Applications: total quantity from density; average value of a 2D function; center of mass

Packet 09

• Partial derivatives: limit definition, trace-curves interpretation
• Definition of tangent plane to graph of function: unique plane containing tangent lines of both trace curves
• Clairaut’s theorem: name, statement
• Linear Approximation Function (LAF)
• Construction of tangent plane using LAF
• Definition of differentiability using LAF and ,
• Differentials notation for doing linear approximation problems, ability to do these problems
• Theorem (statement, not proof): if both partials exist and are continuous, then the function is differentiable

Packet 10

• Derivation of chain rule over a curve using LAF and ,
• Formulation of chain rule for several parameters; application to specific functions
• Directional derivative: limit definition, and gradient-dot formula
• Notation:
  • “in direction of vector …” or “directional derivative of …” = = unit vector needed
  • “derivative of … with respect to vector …” = = any vector allowed
• Gradient: definition; relationship to directional derivative; geometric meaning
• Level curves: definition; derivation that it’s perpendicular to gradient
• Using gradient as normal vector to level surfaces, e.g. to construct tangent plane in vector form

Packet 11

• “Aligned” vector line integrals. (Both notations! i.e. with “ds” and with “… dx … dy … dz.)
• Mastery of 1D optimization techniques. (Local min/max/inflection, 1st and 2nd derivative tests, boundaries, ‘undefined’ critical points, global min/max.)
• 2D optimization; regular vs. critical points; checking boundary curves using 1D optimization
• Finding critical points using gradient / all partials; solving systems of equations
• Ability to write down discriminant test procedure in general rules
• Analyzing critical points using discriminant test
• Analyzing situation with boundary curves (with or without corners), interior critical points, discriminant test, etc., to find global max/min values
• Lagrange multipliers: use in practice for constrained optimization
• Lagrange multipliers: derivation that “gradients align at extremal values”

Packet 13

• Curl: definition as derivative operator
• Curl as measure of circulation: complete derivation for circles (including limit)
• Full statement of Green’s Theorem (including: what is the orientation on !)
• Derivation of Green’s Theorem [not examinable]
• Ability to apply Green’s Theorem. At a minimum:
  • Compute area with line integral over boundary path
  • Easier computation of some line integrals using area integral instead
  • Derive path independence from in simply-connected region

Packet 14

• Know these equivalent properties of a conservative field when the region is simply connected:
  • exists with path independence
• Know the derivations of the above line going left to right
• Fundamental Theorem of Line Integrals: Statement, Derivation, Application (e.g. total change in elevation)
• “Across” vector line integrals
• Divergence as differential operator 2D and 3D
• Polar form of divergence
• Divergence as measure of expansion: complete 2D derivation for circles (including limit)
• Full statement of Divergence Theorem in 2D (including: what is the orientation on !)
• “Green’s Theorem Revisited”

Packet 15

• Parametric surfaces: know the formulas for cylinders, spheres, graphs of functions, and planes (for a plane, each component is a linear function)
• Tangent plane for parametric surface: ability to calculate as linear combinations of and , as well as vector form via the normal vector
• Formula for: positive scalar quantity for scalar surface integrals
• Vector form for: vector quantity for vector surface integrals
• Simplified signed form: change-of-coordinates of -plane (same as vector surface integral with .)
• Jacobian formula (same values as when you interpret as for a reparametrization of the -plane)
• Definitions of scalar and vector surface integrals; formulation in terms of a parametrization; ability to use to compute (i) surface area, and (ii) flux across a surface
• Curl formula as full 3D vector
• Stokes’ Theorem. Pay attention to orientations! (The boundary has a specific orientation to make this theorem true. It is determined by the circulations inside near the boundary, where the circulations must follow the right hand rule such that the thumb points along . The direction of these circulations must correspond to the orientation on the boundary for Stokes’ Theorem to be true. Therefore you may have to change your boundary orientation so it flows along with neighboring interior circulations.)
• Path independence in 3D: excluded from final
• Surface independence for curl fields: excluded from final
• Divergence formula in 3D
• Divergence Theorem. Pay attention to orientations! (The boundary has a specific orientation to make this theorem true. It is determined by “expansions” inside near the boundary, where vectors crossing from inside to outside of must align with the normal vector chosen on to make the Divergence Theorem true. Therefore you may have to change your so that it crosses from inside to outside.)

Exercises and Problems to review

(Some Exercises and Problems in the Packets are omitted from the list because they are more difficult or confusing. Most are included, though.)

Exercises

• 02A-02,-05,-06, 02B-03,-04,-06
• 03A-02,-03,-04,-05, 03B-01,-04,-05,-06,-07, 03C-01,-02,-03,-04
• 04A-01,-02,-04
• 05A-01,-02,-05, 05B-01,-03,-04,-05
• 06A-01,-02,-03,-04,-05,-06,-07; 06B-01,-02,-03,-04
• 08B-all
• 09A-01,-02,-03,-04, B-01,-02,-03,-04
• 10A-01,-02,-03,-04,-05; B-01,-02,-03
• 11A-01,-02,-04, B-01,-02,-03
• 13A-01,-02,-03,-04, B-01,-02,-03
• 14A-01,-02
• 15A-01,-03, B-01,-02,-03

Problems

• (none of 02-xx)
• 03-01, 03-03
• 04-02
• 05-01, 05-03
• (none of 06-xx)
• 08-01(abc), 08-02, 08-03, 08-04(a)
• 09-01(abc), 09-02, 09-03(ac)
• 10-01, 10-03
• 11-01, 11-02, 11-03
• 13-01, 13-02, 13-04, 13-05
• 14-01, 14-03
• 15-02, 15-03, 15-04