# Course overview

Instructor | Contact |
---|---|

Gennady Uraltsev | 📧^{1}🌐 |

Filippo Mazzoli (TA) | 📧^{1} |

Time | Location |
---|---|

– | Zoom |

– | |

(Recitation) | –Zoom |

## Office Hours

Time | Location | Notes |
---|---|---|

– | Zoom | (Gennady Uraltsev) |

– | Zoom | (Gennady Uraltsev) by appointment on Piazza only |

– | Zoom | (Filippo Mazzoli) |

- Meetings WILL NOT be recorded.
- A OneNote link that allows editing will be provided during Office Hours. If you want to be able to write make sure you have a (free) One-Note account and have downloaded the app or can access One-Note online through a browser. The One-Note notebook will be cleared by the end of the day and the link will be disabled.
- You can share your screen with Zoom to show us your work.

## Links

## Summary

The course is an introduction to calculus of functions of several variables. In lower level calculus classes students have learned how to deal with (real) numbers and functions from \(\R\) to \(\R\). While full of very deep results, that knowledge falls short of providing the necessary tools to deal with most real life scientific application because our world is more correctly described by 2, 3, or 4-dimensional space. On the other hand, in a course that deals with linear algebra students learn how to deal with multidimensional vectors but from an algebra perspective.

Calculus 3 (multivariable calculus) bridges these two notions and shows students how to do computations (integrals, derivatives) on surfaces, volumes (a.k.a solids), and paths in 3-dimensional space.

We cover chapters 12 to 16 of Ste.

We will study the geometry of two and three dimensions, including dot and cross products of vectors. Multivariable differential calculus includes partial derivatives, directional derivatives, the chain rule in several variables, and extrema of functions of several variables including the technique of Lagrange multipliers.

We will then study integration of scalar fields (functions) and vector fields, and integration over curves, surfaces and solids.

The course culminates with the fundamental theorem of calculus in higher dimensions, in various incarnations (Green’s, Stokes’, and the divergence theorems particularly). Students are advised that this course moves very quickly. It is very important to stay abreast of the current topics so as not to fall behind. Also note that while the material at the beginning of the course is comparatively elementary, the later topics will be entirely novel and require substantial effort to master.

By the end of this course, you will be able to:

- Describe basic 3-dimensional objects, such as curves, surfaces, solids, and vector fields in various coordinate systems.
- Understand further the concept of differentiation, and adapt its algebraic and geometric interpretations to models in physics, chemistry, economics, and other disciplines you learned or are learning in other courses.
- Explain the mathematical meaning of infinitesimal and infinity, and implement these to study properties of 2- and 3-dimensional objects such as length, area, volume, etc.
- Relate differentiation and integration in various settings and explore how this can give us insights into real-world applications.
- Make concise mathematical arguments about the concepts of the course.

### Question that the course answers

We will develop the knowledge to answer the questions:

- How do I describe a surface in 3D using mathematical formulae?
- Suppose I have a topographical map (describes elevation of terrain); how do I correctly describe the slope of the mountain that I see on the map?
- What are those closed lines (contour lines) on a topographical map?
- I know how to find the maximum value of a function on \(\R\) using derivatives. What if my function describes the temperature at every point in 3D space: what should I use instead of a derivative to find the place where the temperature is highest?
- What if my function is only defined on a surface e.g. I know what the temperature at each point on the surface of the Earth (that is round, b.t.w.) is? How do I use derivatives to figure out which is the coldest place on Earth?
- How much work against gravity is it to climb a given mountain? Does it depend on the path one takes?
- What is the area of the surface of the Earth? What is it's volume? Where do the formulae: \(\frac{4}{3}\pi r^3\) for the volume of a ball and \(4 \pi r^2\) for the surface area of a sphere come from?
- I know Calculus 1: it seems that if I take the derivative of the first formula I obtain the second one. Is this a coincidence? Are there any such cool relations for other shapes?
- Someone described a polygon on the plane to me by giving me the \((x,y)\) coordinates of each corner. I need to compute the area of the polygon, can I do it quickly or do I have to draw it to understand how the shape looks like? After all the polygon can be convex but it could also be concave like this:
- What if I have to write a computer program to compute that area? Am I completely screwed?
- What if instead of a polygon I was given a polyhedron in 3D?
- Do I need to know this to host my own Minecraft server?
- Is there Minecraft in 4D?
- Why can I take the cross product of two 3D vectors and obtain vector, but if I take the cross product of two 2D vectors I get a number?
- Why can I always take the dot product of two vectors in 2D or in 3D and always get just a number?
- What is a cross product, anyway?

### Prerequisites

- Math 1320 or equivalent.

## Assignments and grading

### Homework

Homework is an integral part of the course and represents a major part of the final grade. It will be assigned weekly. There will both be online HW (WebAssign or similar) with multiple-answer / T-F / numerical questions and written HW that you will have to scan and submit through GradeScope. You sometimes will be asked to submit work justifying your answers to your online HW.

### Examinations

We will have a total of 3 examinations: two midterms and a final exam. The first midterm will be graded before the drop day.

### Grading

This is the **final** evaluation scheme. It is *subject to modifications* but will be finalized in the first couple of weeks of class.

Grading scheme | ||
---|---|---|

Assignment | Percentage | Notes |

HW | 50 | Around 25 Webassign exercises per week |

3-4 problems might have a "show your work" component: you must turn in you written solutions. | ||

1 worst HW assignment will be dropped | ||

Midterm 1 | 10 | |

Midterm 2 (oral) | 10 | |

Final | 15 | |

Other | 15 | Starting from the week of | you will have to make small video presentations about solving a problem.

You will also need to submit a review of your midterm 01. | ||

Other small assignments may be required | ||

Grading is: 100% if completed correctly / 50% if evident lack of effort or significant shortcomings / 0% no submission |

**Tentative** letter grade distribution (updated ) (may be subject to change).

Letter Grade | Percentage |
---|---|

A- – A+ | 90-100 |

B- – B+ | 79-90 |

C- – C+ | 67-78 |

D- – D+ | 55-66 |

## How to succeed (one way, there are many!)

### Questions

**Ask questions!** Raise your (virtual) hand or if we do not notice, interrupt us! If we need to finish a concept before responding, we will tell you (with no offense intended) and address your concern as soon as we can. There are no bad questions. Let us know if you are lost trying to understand. Questions do not have to be specific to a step; examples of good questions are:

- If I had to to solve this problem and didn't know how to do it beforehand, how would I come up with the solution?
- Why does this seemingly simpler approach not work here? (maybe you are onto something: do not trust the instructor to be infallible)
- You are using words that you have not used before, what does this even mean?
- Can you draw a picture?

As long as you ask because you actually want to learn and are respectful of everyone in attendance, you are in the clear.

### Working together

You are encouraged to work together on HW and study together. In the few cases when this is not allowed, we will make it explicit. We realize the difficulty of doing so via remote learning. A good place to start is Piazza. Try out asking a question there or attempting to answer someone else's question. Do not worry if your answer might be incomplete or wrong. We will help you out, clarify doubts. If you feel insecure there is always the possibility of posting/answering anonymously on Piazza. But do consider putting your name on your posts: it makes math a bit more human.

### Visit the Math Collaborative Learning Center

### Office hours

Come to office hours! One could say that office hours are really what distinguishes this course from self-learning: you can watch a bunch of YouTube lecture (that may be better than what we teach), but we can actually listen and answer your questions!

### External material

There are many web resources: Wikipedia, Khan Academy, Google, YouTube animations. Look them up. If you are confused: ask us questions

## Course materials

### Textbooks

### Recorded lectures

All lectures will be recorded and made available online. Office hours WILL NOT be recorded.

### Lecture notebook

I will present the lecture material through a screen share of a screen that I will use as a virtual whiteboard. All the written material will be made available online. I use OneNote to write notes. I will provide a read-only link to the OneNote notebook that you can open in the browser and that will update in real-time during the lecture. Since the screen real-estate is limited, I suggest you keep it open because it allows you to scroll back and consult what I have written previously if you get lost or confused.

### Syllabus

I will provide a detailed per-lecture syllabus in the corresponding section. It will address the topics covered and the corresponding pages from the book. The future lecture topics are tentative and subject to change. Once a given lecture is over, the corresponding entry will be expanded and finalized.

I suggest you review the syllabus after each lesson as a way to review and make sure you caught everything covered in the lecture. You should also look at the syllabus before the beginning of each lesson to have a preliminary idea of the topic we will cover.

## Policies

The value of this course lies not only in the material being taught but in the approaches and perspectives that students have interacting with topics presented. Factors like, social identities, personal circumstances and disabilities, access to specific environments and infrastructure all influence the experience each one of us has of this course. I am committed to giving value to your ideas and contribution, to listening, and to building an environment that rewards everyone for communicating and defending your ideas. I encourage all participants to contribute in the way they best see fit to this goal.

Both instructors and students have to make their best effort to facilitate learning. I encourage you to attend because it is during courses that we can engage with each other and share insights, ideas, and guidance for the material covered. However I also understand that everyone has their own learning style and pace and if you prefer a more independent approach we do not wish to penalize you for it by requiring attendance.

I acknowledge that many forms of discrimination and racism are deeply seated in our society, the system of higher education, and the history of UVA, and I believe that at our actions can either reinforcing or work against the systemic injustices. I commit to listening and holding myself accountable for implementing practices that combat discrimination, and encourage you to help me do so. Students of all racial, gender, immigration status and backgrounds are welcome in this classroom. If your status is impacting your ability to succeed in the course please reach out to me to discuss the situation. I commit to listening, being forthcoming, and finding channels of support in full confidentiality unless required by judicial warrant.

With this in mind please consider the following:

### Flexibility

We will try to make reasonable accommodation for any eventuality that students may encounter. If you have any concerns, do not hesitate to contact the instructor. All lecture material is recorded and available to students. If carrying out the coursework is problematic for whatever reason, please reach out.

### Well-being

This course progresses quickly and at times you might feel lost; this is normal. The instructor is available during office hours both to answer your doubts about the course material and suggest ways to approach studying and coursework.

UVA also provides resources if you are feeling overwhelmed, stressed, or isolated. The Student Health and Wellness Center offers Counseling and Psychological Services (CAPS) for its students; call 434-243-5150 to speak with an on-call counselor and/or schedule an appointment. If you prefer to speak anonymously, you can call Madison House’s HELP Line at any hour of any day: 434-295-TALK. Alternatively, you can call or text the Disaster Distress Helpline (1-800-985-5990, or text TalkWithUs to 66746) to connect with a trained crisis counselor; this is toll free, multilingual, and confidential, available to all residents in the US and its territories.

All students enrolled in Fall 2020 courses, and who have successfully completed a FAFSA for the 2020-2021 academic year, can request funding for expenses related to the disruption of fall campus operations due to the pandemic. For information on CARES Act Student Emergency Funding, Bridge Scholarships, and Emergency Loans, please visit Student Financial Services Operational Updates. You might also be eligible for an Honor Loan.

### Attendance

Attendance is strongly encouraged but not mandatory. You will be informed in advance of any graded in-class work.

### Online etiquette

Please do not use mock/joke names in zoom lectures as it can be a disturbance for the class. Go by the name you commonly go by.

Online meetings require some extra attention on behalf of all participants. Please try to make sure your system doesn't echo and is sufficiently quiet when you are unmuted. We are however understanding of various circumstances that may arise as long as you are acting in good faith. If you are in doubt, please contact us.

When you attend an online meeting, strongly consider turning on your webcam. This provides us with a way to gauge your reaction, understand when we are being confusing etc. Turning on the webcam is a way for YOU to help US do a better job.

### Honor code

The University of Virginia Honor Code applies to this class and is taken seriously. Any honor code violations will be referred to the Honor Committee. Upon submission of each assignments in this class you pledge to abide by the rules of UVA, this course, and the specific assignment. If you have doubts, please ask on Piazza.

You MUST acknowledge any help you received on assignments, even if it is permitted: working with a fellow student, looking up things online, etc. Not acknowledging external help is an Honor violation.

### Accommodation

All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered.

## Important dates ^{2}

- Academic Calendar
- A&S Calendar
**First day of class**:**Last day of class**:**Last day to add / change credit options**:**Last day to drop class**:

# Lectures

This is a preliminary lecture plan. It will be adjusted as we go and expanded once each lesson has been covered. Please check back here often.

New terms are marked with `defn`

i.e. "definition". You should make sure you are not confused by what is meant by a new word. This is very important because the easiest way to get lost and fall behind is not understand the meaning of the words that are being used: math just becomes a foreign language.

Useful properties and statements will be marked with `prop`

(proposition).

Very important and deep statements are marked `thm`

(theorem).

Ideas of why a theorem or proposition works will be marked `proof`

. These are usually not full-blown formal mathematical proof but rather ideas that help understand potentially miracolous and misterious statements.

Examples are marked `exmpl`

that you should be familiar with and fully.

Problems left for you to think about are marked with `prob`

(problem).

## Recitation 1

### Topics

- injective and surjective functions
- description of subsets of the plane in terms of equations in the coordinates

## Lecture 1

### Topics:

- Introduction and overview of the course
- Vectors
`defn`

vectors and vector spaces;`defn`

a coordinate of a vector;`defn`

\(\R\), \(\R^2\), and \(\R^3\);- representing vectors;
`defn`

sum of vectors; graphical interpertation of a sum vectors;`defn`

polar coordinates in \(\R^2\).

- Physical meaning of vectors
- vectors represent position or displacement;
- summing vectors as successive displacement;
`exmpl`

A piece of gum stuck to the wheel on a moving car.

- Functions valued in vector spaces
- Functions from \(\R\) to \(\R^2\) or \(\R^3\)

## Lecture 2

### Review:

`defn`

vectors, addition of vectors, scalar multiplication.`defn`

coordinates, polar coordinates (in \(\R^{2}\)).`defn`

Euclidean lenght of a vector.- Why the formula \(\|\vec{v}\|=(v_x^2+v_y^2+v_y^2)^{1/2}\) works in \(\R^3\).

### Topics:

- The equation of a line passing through the origin in the direction of a vector \(\vec{u}\) (in \(\R^2\) and in \(\R^3\)).
- The equation of a line passing through the point \(\vec{w}\) in the direction of a vector \(\vec{u}\) (in \(\R^2\) and in \(\R^3\)).
- The dot product.
- Computing a dot product in \(\R^{2}\).
- Geometric meaning of a dot product in \(\R^{2}\).
`thm`

The relation between the dot product and and \(\cos(\theta)\).- When is a dot product positive, negative, or zero.
`thm`

Cauchy inequality.- Properties of the dot product: the dot product distributes over sums and scalar multiplication.

- Normalization of a vector, a unit vector.
- Orthogonal projection of a vector onto a line;
- Computing a dot product in \(\R^{3}\), its geometric meaning.
- The angle between two vectors in \(\R^{3}\): geometric meaning and computing it using the dot product.

## Recitation 2

## Lecture 3

### Review

- The dot product.
- Computing a dot product.
- Geometric meaning of a dot product.
`thm`

Cauchy inequality.- Normalizing vectors.
- Projecting a vector on a line in \(\R^{2}\) and \(\R^{3}\).

### Topics:

- The cross product in \(\R^{3}\):
- Computing the cross product algebrically
- Properties of the cross product (angle)
- Geometric meaning of the cross product in \(\R^3\)

- Equation of a plane in \(\R^3\).
- The plane orthogonal to a vector.
- The plane containing two vectors in \(\R^3\).

- The cross product in \(\R^{2}\).
- Computing the cross product algebrically.
- Properties of the cross product (angle).

- Equations of lines and planes
- the equation of a plane given two vectors
- the equation of the plane orthogonal to a vector

### References:

- Ste 12.4
- You may skip determinants.
- The deduction of the form of the cross product on p854 is not important.
- Know by heart Definition 4 p855
- Skip Proof of Theorem 9 (p857): it is not very insightful.
- Be able to reproduce the proof of Theorem 8 (p856): there are no tricks involved, just be confident in expanding and cancelling everything out.
- You can skip property 6 in statement 11 (p 859)

- Ste 12.5
- everything

## Lecture 4

### Topics:

- Vector functions
- when is a vector function continuous
`exmpl`

continuous and discontinuous vector functions

- Paths
`defn`

parameterized paths.`defn`

derivative of a path: the velocity vector.- representing the velocity vector.
`defn`

speed is the norm of the velocity vector.

- Derivation and product rules:
- product (Leibniz) rule: the dot product
- product (Leibniz) rule: the cross product

## Recitation 3

### Projection of vectors over a plane in \(\R^3\). Exercise session on WebAssign, topics:

- Ste 10.2, 13.1 and 13.2:
- plots of curves in \(\R^2\) and \(\R^3\),
- tangent line to a curve in \(\R^2\) or \(\R^3\).

- Additional notes on:
- description of the plane passing through 3 points in \(\R^3\),
- line tangent to a curve at a given point.

## Lecture 5

### Topics:

- Derivation and product rules:
- product (Leibniz) rule: the dot product
- product (Leibniz) rule: the cross product

- The chain rule for paths
- Integrals of vector functions
- The length of a path

- The expression for the derivative of the norm of a vector function

### References:

## Lecture 6

### Topics:

- Arc length
- The derivative of the velocity vector: acceleration
- The derivative of the speed: the tangential component of acceleration
- The derivative of the speed: the normal component of acceleration does not change the speed.
- The chain rule and integrals: the lenght of a curve does not change when reparameterizing time.

### References:

## Recitation 4

### Topics

- reparametrization of a curve by arc-length,
- Frenet-Serre triple of a regular curve in \(\R^3\),
- normal and osculating planes to a curve at a given point.

## Lecture 7

### Topics:

- Further topics about curves:
- The tangent vector (normalized velocity vector)
- The normal vector
- The binormal vector and the frame associated with a path
- Arc lenght parameterization

- Kepler's law
- Review

### References:

## Lecture 8

### Topics:

- Functions of several variables
- The domain of a function of several variables
- Representing a function of \(2\) variables: the graph of a function \(z=f(x,y)\)
- the graph of a function as the surface i.e. the set of points of the form \(\Big[x;y;f(x,y)\Big]\)

- Restricting a function of \(2\) variables to a line: studying a cross section of the graph using techniques from Calc1.
- Continuity of a function of two variables:
`defn`

when is a function of two variables continuous?- pathological examples (continuous along some lines but not continuous).

- Representing a function of \(2\) variables: level sets
- examples of level sets of \(f(x,y)=x^2+y^2\), \(f(x,y)=x^2-y^2\), \(f(x,y)=\Big(1-x^2-y^2\Big)^{1/2}\), \(f(x,y)= x^2\)

- Using Geogebra to plot graphs
- Using Geogebra to graphically visualize level sets.
- Representing a function of \(3\) variables: level sets are surfaces
- A graph of a function of \(2\) variables can always be represented as a level set: \(z=f(x,y)\) becomes \(z-f(x,y)=0\), the level set at level \(0\) of \(h(x,y,z)=-f(x,y)+z\).
- Representing a level set of a function of \(3\) variables as a graph is harder: you need to solve for \(z\).
- Partial derivatives: how to compute them
- Partial derivatives as rates of changes along coordinate directions.
- Partial derivatives as 1D derivatives of functions with frozen coordinates (restrictions to lines).
- Directional derivatives.

## Recitation 5

### Topics

- parametrization of a circle in \(\R^3\) given its center, radius and the plane containing it;
- curves in \(\R^3\) lying on the graph of a 2-variables function.

## Lecture 9

### Topics:

- Tangent planes
- Directional derivatives
- The gradient
- The relation between the gradient, the dot product, and the tangent plane.
- The tangent plane as the plane containing all tangent lines of curves on the surface.

## Lecture 10

### Topics:

- The chain rule for functions \(f\colon \R^2\to \R\) and \(f\colon \R^3\to \R\)
- The rate of change of a function \(f\colon \R^2\to \R\) along a curve

## Lecture 11

### Topics:

- Optimization: finding maxima and minima
- finding extrema using derivatives
`defn`

closed sets`defn`

bounded sets- a continuous function on closed and bounded sets attains max and min

### References:

- Ste Ch 14.5 pp 999 - 1001

## Lecture 12

### Topics:

- Optimization: Critical points
- Studying critical points using graphs and/or level sets.

### References:

- Ste Ch 14.7 pp. 999-1001.

## Lecture 13

### Topics:

- Optimization
- The Hessian:
- motivation
- definition

- positive definite/negative definite symmetric \(2\times2\) matrixes
- the second derivative test

### References:

- Ste Ch 14.7

## Lecture 14

### Topics:

- Lagrange multipliers

### References:

- Ste Ch 14.8 pp. 1011-1016 (skip two constraints).

## Lecture 15

### Topics:

- Functions \(f\colon \R^3\to \R\): \(f(x,y,z)\)
- Gradients of 3D functions: direction, magnitude, geometric and physical interpretation.
- Level sets of 3D functions.
- \(\nabla f \) is normal to level sets \(f(x,y,z)=c\).
- Computing the normal line to a level set of \(f(x,y,z)\) at \((x_0,y_0,z_0)\)
- Computing the tangent plane to a level set of \(f(x,y,z)\) at \((x_0,y_0,z_0)\)

- Studying surfaces: level sets vs graphs
- Solving \(f(x,y,z)=c\) for \(z\) to get \(z=h(x,y)\)
- Computing \(\nabla h\) when knowing \(\nabla f\): \(\nabla h (x,y) = -\frac{1}{\partial_z f(x,y,z)} \begin{pmatrix}\partial_x f(x,y,z)\\ \partial_y f(x,y,z)\end{pmatrix}\)
`thm`

implicit function theorem: if \(f\colon \R^3\to \R\) is differentialble with continuous gradient around \((x_0,y_0,z_0)\) and \(\partial_zf(x_0,y_0,z_0)\neq 0\) then the level set \(f(x,y,z)=f(x_0,y_0,z_0)\) around \((x_0,y_0,z_0)\) can always be described by the graph \(z=h(x,y)\).- Such an \(h\) is differentiable and satifies the relation above.

- Remarks
- Lagrange multipliers work in \(\R^3\), too!

### References:

## Lecture 16

### Topics:

- Integrals in 2D
- Motivation and physical applications: summing up densities, computing volumes
- Relation with definite integrals in 1D
- Review: Riemann sums in 1D
- Riemann sums in 2D: summing over small squares
- Commuting order of summation in 2D Riemann sums: an intuition about Fubini's Theorem

`thm`

Fubini Theorem in 2D over rectangular domains: you can change order of integration- Integrating a function \(f(x,y)\) over a rectangular domain \(R=[a,b]\times[c,d]\)
- Integrating a function \(f(x,y)\) over a non-rectangular domain \(\Omega\): set \(f\) to \(0\) outside \(\Omega\) and enclose \(\Omega\) with a rectangle
- The issue of describing the boundary of \(\Omega\): \(p_-(x)\leq y\leq p_+(x)\) or \(q_-(y)\leq x\leq p_+(y)\): both choice are valid but require you to integrate in a certain order.

## Lecture 17

### Topics:

- Integrals in 2D in polar coordinates
- Motivation, geometric interpretation: summing up densities, computing volumes
- Why is the area of the circle derived in the radius giving you the length of the circumference.

- Integrals in 2D: splitting areas and using different coordinate systems
- Application: Gaussian integrals.
- Improper integrals
- Integrals in 3D
`thm`

Fubini- Integrals over rectangles
- Integrals over general regions in 3D
- Cylindrical coordinates

## Lecture 18

### Topics:

- Change of variables formula
- Spherical coordinates

## Lecture 19

### Topics:

### References:

## Lecture 20

### Topics:

### References:

## Lecture 21

### Topics:

### References:

## Lecture 22

### Topics:

### References:

## Lecture 23

### Topics:

### References:

## Lecture 24

### Topics:

### References:

## Lecture 25

### Topics:

### References:

## Lecture 26

### Topics:

### References:

# Hand-in Homework

## HW03

Assigned: Gradescope

Due: Please submit your assignment on# Midterms and Exams

## Midterm 01

The midterm is going to be an assignment/quiz administered through Collab. It is going to be 90min long.

Exercises will be of similar format to the ones done on Webassign:

- multiple choice
- true/false
- numerical computation
- symbolic computation (e.g. "parameterize this curve")

Some exercises will require you to show your work (details will be provided on the options of submitting your work).
There will **not** be questions like "justify/ show that.."

You will be able to go back to your questions before submitting. The system will, however, **not** tell you whether your answers are correct prior to the final submission.

### Zoom link:

### Rules and Proctoring

With remote learning we are doing our best to provide a fair, productive method of assessment and test taking. By taking this midterm you pledge to the UVA Code of Honor.

We implement measures to enforce academic integrity to guarantee an equal and fair testing environment. We thank you for your patience and understanding. If you have any concerns, please let us know.

- The exam will be proctored via Zoom.
- You will be in individual breakout rooms. We will periodically check in on you.
- You will be asked to share your webcam feed and screen for the duration of the test. Your screen feed must be shared for the whole duration of the exam.
- Make sure your desk is clearly visible and message us via Zoom chat before the start of the exam if you have additional personal notes. We may ask you to show them to us using the webcam. Please tell us if you have PDF notes that you will access using your PC.
- Make sure that all other devices (e.g. phones/tablets) and screens you are not sharing are TURNED OFF.
- Your Zoom video feed is recorded while you are taking the test. This recording will be kept private and deleted immediately after the grading of the midterm is finished.
- Please make sure to close all external programs for the duration of the test. Using your browser you may: access WebAssign (e-book only), access Collab to take the test, access Collab to access lectures, access one-note lecture notes, and an online non-graphing calculator.

Exam format:

- The exam has multiple parts. They are all visible simultaneously on the same webpage.
- You can save your answers, if you do not submit by the end of the allowed time, your last saved result will be your submission.
- The questions may be: T/F, multiple choice, numerical answer, and free form answer.
- For numerical answer questions the precision of your answer is specified. You may use the following online non-graphing calculator to compute your results
- Free-form answers will be graded by humans. Any typesetting of your answer is valid as long as we can understand it. If you know how to use LaTeX you may, but this is NOT MANDATORY
- Please take your time now to think how you would write up an integral \(\int_a^b cos(t) dt\) a vector \([1;2;3]\) or a square root \(\sqrt{x^2+y^2+1}\).
- Have paper and pencil ready.
- We will open a gradescope submission where to submit a PDF of your work. We will consider your work for partial credit if your response online is incorrect.
- Your response online is the one trusted. If it is wrong, partial credit will be awarded only if your work (solution) corresponds to the WRONG answer provided and we can identify an honest mistake. Submitting your work is NOT a way to have two attempts at solving a problem.
- You will have time after the midterm submission to scan/photo your work and submit it on gradescope.
- While doing so you still have to be connected to Zoom and have screen-sharing / webcam enabled.

A tentative meeting to make sure that your system is set up correctly will be held

. If you have any concerns about your ability to be proctored during the scheduled time, let us know ASAP.- The exam is open book (e-book). You may log in to Webassign and consult your book.
- You may consult the lecture notes/recitation notes.
- You may consult personal notes, however you
**must**let us know what you are bringing (via private message on Zoom prior to the start of the exam). - Any other materials or forms of external help are prohibited.
- In particular, graphing calculators/Desmos/Geogebra are
**not**allowed.

### List of topics

All topics covered until lesson

, inclusively, may be on the exam.A large list of things you should know how to do and what they are. For every one of these, make sure you know WHAT the referenced concepts are, examples where they are used, and examples of how to carry out corresponding computations.

- Parametric equations, cartesion coordinates, polar coordinates, and vectors, equations, and parameterizations
- Parameterize a line on a plane and in space given a point and a direction
- Parameterize uniform circular motion in a plane given center, radius, initial angle, and angular velocity
- Parameterize an ellipse
- Parameterize a curve lying on a graph of a function \(f\colon \R^2\to \R\).
- Write a line in \(2D\) in equation form
- Write a line in \(3D\) in equation form (2 equations)
- Parameterize a plane in \(2D\) given a point and a normal vector
- Parameterize a plane in \(2D\) given a point and two direction vectors
- Write a plane in \(3D\) in equation form (1 equation)
- Pass freely from from equation to parametric form for linear objects (lines/planes)

- Dot, cross products, projections, normal vectors, direction vectors
- Compute dot products (using coordinates and using geometry)
- Compute cross products (using coordinates and using geometry)
- Properties of dot products
- Properties of cross products
- Compute a projection of a point onto a line (the line expressed either in parametric or equation form)
- Compute a projection of a point onto a plane (the line expressed either in parametric or equation form)
- Find the orthogonal direction in \(\R^2\) to a vector
- Find the orthogonal direction in \(\R^3\) to a plane or 2 vectors
- Find the orthogonal directions in \(\R^3\) to a vector
- Finding parallel directions and orthogonal directions of planes or lines expressed in parametric or equation form
- Computing angles between 2 lines, 2 planes, or line and plane.

- Vector functions and paths
- Parameterize a path in \(2D\) or \(3D\)
- Compute a path's velocity, speed, acceleration
- Compute a path's length
- Compute the tangent line to a path at a point
- Say whether a path is arch-length parameterized
- Determine tangent, normal, binormal directions of an arch-length parameterized path
- Determine tangent, normal, binormal directions of a NON arch-length parameterized path
- Chain rule for paths: compute the derivative of \(f(\phi(t))\)
- Chain rule for paths: reparameterizing time
- Derivatives of products: Leibniz rules
- The formula of the derivatives of the length of a vector
- The formula of the derivatives of the length of a vector: apply to find the rate of change of speed
- The formula of the derivatives of the length of a vector: apply to find the rate of change of the distance to a fixed point
- The role of tangential acceleration: changing speed
- The role of normal acceleration: changing direction

- Representing functions \(f\colon \R^2 \to \R\)
- Examples of functions \(f\colon \R^2 \to \R\): paraboloid, cone, hemisphere, ellypsoid, saddle (hyperboloid)
- Graph of a function
- Graphs of examples of functions \(f\colon \R^2 \to \R\): paraboloid, cone, hemisphere, ellypsoid, saddle (hyperboloid)
- Level curves of a function \(f\colon \R^2 \to \R\): possible shapes

- Multivariable functions
- Find the domain of a function
- Decide whether a function is continuous or not (no proof)
- Decide whether a function is differentiable or not (no proof)
- Find the tangent plane to a graph of a function \(f\colon\R^2\to\R\)
- Find the normal line to a graph of a function \(f\colon\R^2\to\R\)
- Find tangent line to the level curve of a function \(f\colon\R^2\to\R\)
- Check whether a curve lies on the graph of a function
- Tangent directions to the graph as all possible tangent directions to paths lying on the graph

- Gradients and partial derivatives
- Find the partial derivatives of a function
- Geometric meaning of the partial derivatives of a function
- Compute the gradient
- Geometric meaning of the gradient: the direction of a gradient
- Geometric meaning of the gradient: the magnitude of a gradient
- Relation of the gradient with level curves: direction
- Relation of the gradient with level curves: density of level curves and magnitude of the gradient
- Apply the chain rule to \(f(p(t))\) given \(f\colon \R^2\to\R\) and \(p\colon\R\to \R^2\)
- Product rule for derivative \(f(x,y)g(x,y)\).
- Fundamental thm of calculus: Integrate the gradient along a curve.
- Compute the directional derivative
- Find the tangent plane to a graph given the gradient of a function
- Find the normal line to a graph given the gradient of a function
- Find the tangent line to a level set given the gradient of a function
- Draw level curves freehand given the gradient
- Draw the gradient freehand given level curves

- Functions \(\R^3\to \R\)
- Level surfaces of a function \(f\colon\R^3\to \R\)
- Examples level surfaces of a function \(f\colon\R^3\to \R\): a sphere, a cylinder, a cone, an ellipsoid
- Find the tangent plane to the level surface of a function \(f\colon \R^3\to \R\)

- References:
- Ste Ch 10: Parametric equations and polar coordinates
- Ste Ch 12: Vectors and the geometry of space
- Ste Ch 12.0: Introduction (everything)
- Ste Ch 12.1: Three-Dimensional Coordinate Systems (everything)
- Ste Ch 12.2: Vectors (everything)
- Ste Ch 12.3: The Dot Product (everything)
- Ste Ch 12.4: The Cross Product (everything except torque)
- Ste Ch 12.5: Equations of lines and Planes (everything)
- Ste Ch 12.6: Cylinders and Quadric Surfaces (basics - not in detail)

- Ste Ch 13: Vector functions
- Ste Ch 13.0: Introduction (everything)
- Ste Ch 13.1: Vector Functinos and Space Curves (everything)
- Ste Ch 13.2: Derivatives and Integrals of Vector Functions (everything)
- Ste Ch 13.3: Arc Length and Curvature (everything)
- Ste Ch 13.4: Motion in Space: Velocity and acceleration (everything, except projectile motion)

- Ste Ch 14: Partial Derivatives
- Ste Ch 14.0: Introduction (everything)
- Ste Ch 14.1: Functions of Several variables (everything)
- Ste Ch 14.2: Limits and Continuity (everything)
- Ste Ch 14.3: Partial Derivatives (everything)
- Ste Ch 14.4: Tangent planes and Linear Approximations (everything)
- Ste Ch 14.5: The chain rule (everything)
- Ste Ch 14.6: Directional Derivattives and the Gradient Vector (everything)

## Midterm 02

### Time (by appointment)

- –
- –
- -
- –
- -

### Format:

- Closed book oral exam (approx 20min) on Zoom.
- 2-3 Questions.
- Please make sure you have good equipment to be able to show me your work.

### Assessment

The oral exam will be assessed with a percentage grade. You will be asked to show your intermediate work. However you will have some time to work through your computation before starting you answer.

### List of topics

- Ste Ch 14: Partial Derivatives
- Ste Ch 14.0: Introduction (everything)
- Ste Ch 14.1: Functions of Several variables (everything)
- Ste Ch 14.2: Limits and Continuity (everything)
- Ste Ch 14.3: Partial Derivatives (everything)
- Ste Ch 14.4: Tangent planes and Linear Approximations (everything)
- Ste Ch 14.5: The chain rule (everything)
- Ste Ch 14.6: Directional Derivattives and the Gradient Vector (everything)
- Ste Ch 14.7: Maximum and Minimum Values (everything)
- Ste Ch 14.8: Lagrange Multipliers (one constraint)
- Implicit function theorem (done in class)
- Level sets / gradients in 3D.

- Ste Ch 15: Multiple Integrals
- Ste Ch 15.0: Introduction (everything)
- Ste Ch 15.1: Double integrals over rectangles (everything)
- Ste Ch 15.2: Double integrals over general regions (everything)
- Ste Ch 15.3: Double integrals in polar coordinates (everything)
- Ste Ch 15.6: Triple integrals (everything)
- Ste Ch 15.7: Triple integral in cylindrical coordinates (everything)
- Ste Ch 15.8: Triple integrals in spherical coordinates (one constraint)

## Final

### Zoom link:

Please log in 15 min before the start of the exam.

### Rules and Proctoring

With remote learning we are doing our best to provide a fair, productive method of assessment and test taking. By taking this midterm you pledge to the UVA Code of Honor.

We implement measures to enforce academic integrity to guarantee an equal and fair testing environment. We thank you for your patience and understanding. If you have any concerns, please let us know.

- The exam will be proctored via Zoom.
- You will be in individual breakout rooms. We will periodically check in on you.
- You will be asked to share your webcam feed and screen for the duration of the test. Your screen feed must be shared for the whole duration of the exam.
- Make sure your desk is clearly visible and message us via Zoom chat before the start of the exam if you have additional personal notes. We may ask you to show them to us using the webcam. Please tell us if you have PDF notes that you will access using your PC.
- Make sure that all other devices (e.g. phones/tablets) and screens you are not sharing are TURNED OFF.
- Your Zoom video feed is recorded while you are taking the test. This recording will be kept private and deleted immediately after the grading of the midterm is finished.
- Please make sure to close all external programs for the duration of the test. Using your browser you may: access WebAssign (e-book only), access Collab to take the test, access Collab to access lectures, access one-note lecture notes, and an online non-graphing calculator.

Exam format:

- The exam has multiple parts. They are all visible simultaneously on the same webpage.
- You can save your answers, if you do not submit by the end of the allowed time, your last saved result will be your submission.
- The questions may be: T/F, multiple choice, numerical answer, and free form answer.
- For numerical answer questions the precision of your answer is specified. You may use the following online non-graphing calculator to compute your results
- Free-form answers will be graded by humans. Any typesetting of your answer is valid as long as we can understand it. If you know how to use LaTeX you may, but this is NOT MANDATORY
- Have paper and pencil ready.
- Please take your time now to think how you would write up an integral \(\int_a^b cos(t) dt\) a vector \([1;2;3]\) or a square root \(\sqrt{x^2+y^2+1}\).
- We will open a gradescope submission where to submit a PDF of your work. We will consider your work for partial credit if your response online is incorrect.
- Your response online is the one trusted. If it is wrong, partial credit will be awarded only if your work (solution) corresponds to the WRONG answer provided and we can identify an honest mistake. Submitting your work is NOT a way to have two attempts at solving a problem.
- You will have time after the midterm submission to scan/photo your work and submit it on gradescope.
- While doing so you still have to be connected to Zoom and have screen-sharing / webcam enabled.

If you have any concerns about your ability to be proctored during the scheduled time, let us know ASAP.

- The exam is open book (e-book). You may log in to Webassign and consult your book.
- You may consult the lecture notes/recitation notes.
- You may consult personal notes, however you
**must**let us know what you are bringing (via private message on Zoom prior to the start of the exam). - Any other materials or forms of external help are prohibited.
- In particular, graphing calculators/Desmos/Geogebra are
**not**allowed.

### List of topics

- Physical interpretation
- Invariance w.r.t to direction-preserving reparametrisation
- Relation between direction and sign of the path integral
- Concatenation properties
- Examples of paths: A path with constant radius, a piecewise-linear path
- Length of a path.
- Application of Cauchy-Schwarz to path integrals: A path integral is smaller than the length of the path times the maximum magnituge of the vector field along the path
- Closed loops
- Line integrals over closed loops: does not depend on starting point of parameterization
- Non self-intersecting paths

- The integral along a path of a gradient vector field does not depend on the specific path but only on the beginning and end points of the path.
- Application: path integrals of gradient vector fields over closed loops.
`defn`

: Conservative vector fields- examples of most common conservative vector fields (see Vector fields in 2D and 3D)
`thm`

: A continuous vector field is conservative if and only if path integrals depend only on starting and end points (2D and 3D)- path integrals over two paths that start at the same place and end up in the same place: relation with path integrals over closed loops (2D and 3D)
`thm`

: A continuous vector field is conservative if and only if all path integrals over closed loops are \(0\) (2D and 3D)- Given a vector field \(\vec{F}\) using the fundamental theorem of calculus to find the potential \(f\) ( \(\nabla f = \vec{F}\)): integrating along paths
- Given a vector field \(\vec{F}\) guessing the potential \(f\) ( \(\nabla f = \vec{F}\))
- Exploiting symmetries

- Connected domains of \(\R^{2}\)
- Simply connected domains of \(\R^{2}\)
- Boundaries of domains in \(\R^{2}\): describing boundaries of domains in \(\R^{2}\) using paths
- Domains with boundaries that have multiple separate parts: domains in \(\R^{2}\) with holes
- Normal vector to the boundary of a domain in \(\R^{2}\)
- Orienting the boundary of a domain in \(\R^{2}\)

`defn`

Parameterized surfaces in \(\R^{2}\)- Examples: paraboloid, part of sphere, side of cylinder, flat \(2D\) shapes in \(\R^{3}\), torus
- Examples: parameterizations of surfaces
- Orientation and orientable surfaces: examples and counterexamples
- Choosing a parameterization with the correct orientation
- normal vector to a parameterized surface
- the unit of infinitesimal area of a parameterized surface: \(\|\partial_{u} \vec{s}(u,v)\times \partial_{v} \vec{s}(u,v)\|\).
- Computing the area of a parameterized surface

- Connected domains of \(\R^{3}\)
- Simply connected domains of \(\R^{3}\)
- Boundaries of domains in \(\R^{3}\): describing boundaries of domains in \(\R^{3}\) using parameterized surfaces
- Domains with boundaries that have multiple separate parts: domains in \(\R^{3}\) with holes
- Orienting the boundary of a domain in \(\R^{3}\)

- The flow of a field through a path in \(\R^{2}\)
- The flow of a field through a surface in \(\R^{3}\)

`defn`

The operators \(\mathrm{curl}\) and \(\mathrm{div}\) in \(\R^{2}\)`defn`

The operators \(\mathrm{curl}\) and \(\mathrm{div}\) in \(\R^{3}\)- What kind of objects are \(\mathrm{curl}\) and \(\mathrm{div}\) in \(\R^{2}\) and in \(\R^{3}\). Relation between \(\mathrm{curl}\) in \(\R^{2}\) and \(\mathrm{curl}\) in \(\R^{3}\)
- Physical meaning of \(\mathrm{curl}\) and \(\mathrm{div}\).
- Algebraic identities: \(\mathrm{curl}\nabla=0\) \(\mathrm{div}\mathrm{curl}=0\).

- Stokes for \(\mathrm{curl}\) in \(\R^{2}\)
- Non-intersecting closed curves, circuitation and oriented boundary
- Dealing with domains with holes: adding paths and cutting domains.
- Fundamental theorem of calculus for line integrals: if the domain \(\Omega\subset\R^{2}\) is simply connected and \(\mathrm{curl} F=0\) then \(F\) is conservative
- The curl and deforming paths
- Noteworthy fields \(F(x,y)=(x,y)\), \(F(x,y)=(-y,x)\), \(F(x,y)=\frac{1}{\|(x,y)\|^{2}}(x,y)\) and their curls

- Stokes for \(\mathrm{div}\) in \(\R^{2}\)
- Orientation of the boundary. The normal vector
- Dealing with domains with holes.
- Noteworthy fields \(F(x,y)=(x,y)\),\(F(x,y)=\frac{1}{\|(x,y)\|^{2}}(x,y)\), \(F(x,y)=(-y,x)\), \(F(x,y)=\frac{1}{\|(x,y)\|^{2}}(-y,x)\) and their \(\mathrm{curl}\)

- Stokes for \(\mathrm{div}\) in \(\R^{3}\)
- The outward normal
- Dealing with domains with holes.
- Noteworthy fields \(F(x,y,z)=(x,y,z)\),\(F(x,y,z)=\frac{1}{\|(x,y,z)\|^{3}}(x,y,z)\) and their \(\mathrm{div}\).

- Stokes for \(\mathrm{curl}\) in \(\R^{3}\)
- The flow of the curl
- Orienting the boundary of a parameterized surface
- Dealing with surfaces with holes
- Noteworthy fields \(F(x,y,z)=(x,y,z)\),\(F(x,y,z)=\frac{1}{\|(x,y,z)\|^{3}}(x,y,z)\) and their \(\mathrm{curl}\)
- The rotation field: \(F(x,y,z) = (a,b,c)\times (x,y,z)\) for some fixed \((a,b,c)\). Its \(\mathrm{curl}\)
- Given a closed curve in \(\R^{3}\) the liberty of choosing a surface with that boundary
- The flow of the \(\mathrm{curl}\) through a surface enlosing a volume

# Presentations

Please submit under "File Drop" in Collab.

## Presentation01

Assigned:

Due:## Presentation02

Assigned:

Due:## Presentation03

Assigned:

Due:## Presentation04

Assigned:

Due:# Extra problems and questions

## Footnotes:

^{1}

Do not use e-mail for any questions about course topics. All course material should be discussed on Piazza, e-mails about course topics will be ignored WITHOUT WARNING.

^{2}

These dates are provided for reference only. Please consult the official UVA registrar.