MA 242 Review Topics
This outline of the course topics is meant to serve as a guide for what topics to study as you prepare for the final exam. We recommend that you look back at your notes, the videos, Webassign, worksheets, and textbook to review each topic and remember/summarize what the key formulas, skills, and concepts were. You should plan on practicing with both topics that you think you know well, and ones that you don’t. You are allowed one page of notes on the final exam, so this outline may guide you in making that.
1. Unit 1: Geometry of R3
(a) Getting used to R3
i. Distances in R3
ii. Translating among pictures, descriptions, and equations for cylinders, planes, and spheres
in R3. (Could you draw a picture and write down an equation for the cylinder of radius 3 centered around the line where x = 5 and z = 2. Does this cylinder touch the xy-plane? the x-axis? If not, how far away is it from each?
iii. Vectors: vector addition, scalar multiplication, magnitude, using ˆi, ˆj, kˆ notation, using vectors to model force.
(b) Dot and cross products
i. Dot product: algebraic and geometric definitions, using dot product to find angles, how to tell if two vectors are perpendicular, application to work
ii. Vector projection: formula, geometric interpretation (can you draw a picture of this? what happens when the vectors form an obtuse angle?)
iii. Cross product: how to compute, geometric characterization (magnitude is ?, direction is ?), application to torque
iv. When is dot product positive? negative? zero? When is cross product zero?
(c) Lines and Planes
i. How to parameterize a line
ii. How to write down the equation for a plane
iii. Remember all of the problems about finding the distances and angles between lines and planes, finding the closest point along a line or plane to another, etc. Practice with these kinds of problems!
2. Unit 2: Curves in R2 and R3
(a) How to use a vector-valued function to: a) describe a curve in R3, b) describe the position of an
object as a function of time.
(b) Position, velocity, speed, acceleration
(c) Unit tangent vector and curvature
(d) Arc length
3. Unit 3: Multivariable functions and differential properties
(a) Multivariable functions
i. What is meant by the “graph” of a function of two variables? Give some examples.
ii. What is meant by the “level sets” or “contour plot” of a function of two variables? three variables? Give some examples.
iii. Look at some examples of graphs and contour plots. What were the kinds of questions we asked you about these early on? What features can you identify from each kind of visualiza- tion?
iv. Why do we never ask about the graph of a function of three variables? 1
(b) Quadric surfaces
i. If you are given a quadric equation in x,y,z, how can you identify which kind of surface it defines? How can you use cross-sections to help?
ii. Some quadric surfaces are graphs of functions and some are not. Which are?
(c) Partial Derivatives
i. How to calculate them
ii. How to identify and visualize them geometrically from a graph or contour plot
iii. Directional derivatives: how to visualize and how to calculate with dot product
iv. Clairaut’s theorem
(d) Linearization
i. What is the linearization of a multivariable function? Why it is useful? ii. How to estimate values of a function from a linearization.
iii. How is linearization related to tangent planes?
(e) Gradient and applications
i. Make a list of everything you need to know about the gradient1
(f) Optimization
i. What are critical points? How to find them. How to tell if they are local maxes, mins, or neither.
ii. What is the difference between a local max and a global/absolute max?
iii. How would you find the global/absolute max and min of a function f(x,y) on the region in R2 where 0 ≤ y ≤ 1 − x2. Where would you need to check? How would you check? Are you guaranteed to actually have a global max and min on this region or is it possible that they
don’t exist?
4. Unit 4: Double and triple integrals
(a) Double integrals
i. Give an outline of the definition of a double integral of a function f(x,y) on a region R in R2.
ii. What are some examples of applications of double integrals? How does each of these work?2
iii. How do you find the limits of integration when converting a double integral into an iterated integral? How do you decide what order to write the integrals in? (Look back at some early
examples of this for practice.)
(b) Triple integrals
i. Same bullet points as double integrals
5. Unit 5: Double and triple integrals using other coordinate systems
(a) What are the conversion formulas from polar to cartesian? cylindrical to cartesian? spherical to cylindrical? spherical to cartesian?
(b) Draw pictures of what the polar variables represent. Do the same for cylindrical and spherical.
(c) What are some easy-to-describe surfaces or shapes in each of the coordinate systems? Give examples.
(d) What things do you need to keep in mind when converting a double/triple integral into a coordi- nate system?3
1Ideas: how to calculate, relationship to level sets, how to use gradient to find tangent planes, how to use gradient to find directional derivatives, multivariable chain rule, relationship to linearization, points in the direction of fastest increase. . .
2Ideas: totaling a density function, area, average value, volume under the graph…
3Three things: convert the function, convert the bounds, convert the differential—what are the formulas for dA and dV in your coordinate systems?
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(e) Do a bunch of examples!
6. Unit 6: Line and surface integrals
(a) Vector fields
i. How do we visualize a vector field? Can you sketch a simple one like ⟨y, x⟩?
ii. What are some applications of vector fields?4
iii. What does it mean for a vector field to be conservative? What is a potential function? How
can you tell if a vector field is conservative or not? How can you find a potential function?
(b) Line integrals (of vector fields), aka work integrals
F⃗ · d⃗r. Could you tell from a picture
ii. How do you calculate a line integral?
iii. What does the fundamental theorem of line integrals say? Why does it work? Under what
circumstances does it give you a shortcut for calculating line integrals? Under what circum- stances doesn’t it?
(c) Parameterizing surfaces: outline a few techniques for parameterizing surfaces. How would you do it for a surface like y = x3 + sin z with 0 ≤ x ≤ 2 and 0 ≤ z ≤ 3? a surface like x2 + y2 = z3 + z with0≤z≤3? asurfacelikex2+y2+z2 =9withx≥0,y≥0andz≥0? Includethe parameter domains.
(d) Surface area: how do you calculate it? Why does ∥⃗ru ×⃗rv∥ show up in the formula?
(e) Flux
i. Describe geometrically what flux measures about the relationship between a surface and a vector field
ii. What is meant by “orientation” of a surface? How does this manifest in calculating flux integrals? Where should you be careful about orientation?
iii. How do you calculate a flux integral F⃗ · dS⃗? Why does ⃗ru × ⃗rv show up in the formula? U
iv. Suppose that for a surface U with normal vector field nˆ and vector field F⃗ we know that F⃗ · nˆ = 3 is constant and that the surface area of U is 7. What is the flux of F⃗ through U?5
7. Unit 7: Vector Analysis
(a) Divergence, Curl, and 2D: how to calculate them? how to identify/characterize them geometri- cally?
(b) Green’s Theorem
i. What does Green’s Theorem say?6
ii. Give an example of a line integral that can be converted to a double integral using Green’s Theorem.
iii. How can you use Green’s Theorem to find area?
(c) Stokes’ Theorem
i. What does Stokes’ Theorem say?7
ii. Give an example of a line integral that can be converted to a flux integral using Stokes’
Theorem.
4Ideas: velocity of a fluid, force, gradient. . . 5Hint:remembertheformula F⃗·dS⃗= F⃗·nˆ∥⃗ru×⃗rv∥dudv
6Things to be careful of: relationship between the region R and the curve C, orientation, equation, can you draw a picture of what’s going on?
7Things to be careful of: relationship between the surface S and the curve C, orientation, equation, can you draw a picture of what’s going on?
i. Describe geometrically what is meant by the notation whether this would be positive or negative?
C
UD
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iii. Stokes’ Theorem can sometimes be used to convert a flux integral into a line integral, but not for every flux integral. Give an example of a flux integral to which Stokes’ Theorem does apply, and one to which it does not.8
(d) Divergence Theorem
i. What does Divergence Theorem say?9
ii. Give an example of a flux integral that can be converted to a triple integral using Divergence Theorem.
iii. Divergence Theorem can sometimes be used to convert a flux integral into a triple integral, but not for every flux integral. Give an example of a flux integral to which Divergence Theorem does apply, and one to which it does not.10
iv. How can you use Divergence Theorem to find volume?
8. Overarching concepts. Before the final, it’s useful to reorganize your thinking about certain topics. For example, we’ve done a lot with line integrals, but it was spread out over several units. On the final, you might see a question about line integrals, and you’ll need to decide which of the things you know about it is relevant. Here are some ideas for getting yourself well-prepared.
(a) You’ve learned a lot about gradient over the course of the semester. Make a list of everything you know about gradient now: its applications and properties in particular.
(b) You’ve learned a lot about tangent planes over the course of the semester. Make a list of everything you know about tangent planes now: techniques for calculating them in particular.
(c) You’ve learned a lot about line integrals over the course of the semester. Make a list of everything you know about line integrals now: techniques for calculating it in particular.
(d) You’ve learned a lot about flux integrals over the course of the semester. Make a list of everything you know about flux integrals now: techniques for calculating it in particular.
(e) You’ve learned a lot about equations of surfaces over the course of the semester. Make a list of everything you know about equations of surfaces now: techniques and examples that you want to remember in particular.
(f) You’ve learned a lot about parameterizing surfaces over the course of the semester. Make a list of everything you know about parameterizing surfaces now: techniques and examples that you want to remember in particular.
(g) We’ve used contour plots to visualize a lot of properties of multivariable functions over the course of the semester. Make a list of everything you know about contour plots now: how to visualize different properties of their functions in particular.
8Hint: Stokes’ Theorem is about flux-of-curl.
9Things to be careful of: relationship between the surface S and the solid E, orientation, equation, can you draw a picture of what’s going on?
10Hint: What is the role of E?
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