$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Study Guide I

Write a study guide for yourself in preparation for Midterm I.
You can write the guide however helps you most, but here are some possible suggestions:

Write down the important definitions for each topic you want to review. Explain them in your own words!
Either come up with an example problem (or find one on WebAssign) for each topic you want to review, and write out its solution by hand! You’ll be handwriting soltuions on the exam, so this is a good way to get yourself in that mindset and away from online work.
For each topic you want to review, write to yourself how confident you are with it (maybe on like a 1-10 scale, if that’s helpful to you): this way when you’re studying you can look back and easily remember which things you were struggling with.
If there’s multiple ways to do something, make a note of that for yourself!

Below I’ve made a short list of the topcis we’ve covered so far which could appear on the exam. (This is essentially a table of contents for the first two sections of our notes, but will be helpful to those who are primarily reading the textbook instead). I will ensure test problems only cover the below topics.

Space

Dimensions

What is the dimension of a space? Can you calculate the dimension of various example spaces (surface of the earth, a space of images, configuration space of a robot, etc).
Be comfortable using cartesian coordinates and $n$-tuples to represent points in space. Understand polar and cylindrical coordinates as alternative methods to represent points using $n$-tuples.
Be able to use the distance function in $\RR^n$.

Vectors

Know the difference between a point (location) and a vector (direction + magnitude). Be able to give examples of quantities that are points versus vectors.
Be able to use vector addition and scalar multiplication. Understand how they work geometrically as well as in terms of cartesian coordinates.
Be able to compute the magnitude of a vector, find the unit vector in a given direction, and combining these - find a vector of a given length in the direction of another vector.
Be familair with the two notations for vectors using $n$-tuples $\langle x,y,z\rangle$ as well as $ijk$ notation $x\ihat+y\jhat+z\khat$.

Operations

Dot Product

Know the formula for the dot product, and be able to use it. Know how the dot product interacts with vector addition and scalar multiplication.
Understand the relationship between the dot product and angles. Be able to find the angle between two vectors (you will not have a calculator, so it is fine if your answer looks like $\arccos(3/4)$ or something on the test.) Know the definition of orthogonality, and how to test if two vectors are orthogonal with the dot product.
Be able to compute the scalar and vector projections of one vector onto another.

Cross Product

Know how to compute the cross product using determinants. Be able to compute the cross product of the standard basis vectors $\ihat,\jhat,\khat$ using the “circle” diagram.
Understand the geometric meaning of the cross product, the right hand rule, and the connection between the length of the cross product and the area of a parallelogram.
Be able to use the triple produce $a\cdot (b\times c)$ to calculate the volume of a 3D parallelpiped.

Shapes

Lines and Planes

Be comfortable computing with 2D lines given in the form $ax+by=c$.
Be comfortable computing with 3D planes given in the form $ax+by+cz=d$.
Understand the relationship between normal vectors to a line/plane and the vector of coefficients: how did we derive this again?
Be able to find the normal vector to a plane given points on the plane, or vectors parallel to the plane, using the cross product.
Be able to tell when two lines or planes are parallel, or when a line is parallel to a plane in 3D.
Find the angle of intersection between two lines or planes.

Quadratic Curves and Surfaces

Be able to recognize basic 2D quadratic curves from their equations (parabolas, circles, ellipses, hyperbolas).
Understand the relationship between the equation for a circle in 2D and a cylinder in 3D.
Given a quadratic surface in 3D, be able to draw slices along the $xy, yz$ or $xz$ planes to get a sense of its shape.
Be able to draw slices along a sequence of parallel planes (like $z=-1,z=0$ and $z=1$) to get a sense of the overall shape by ‘stacking’.
Find points of intersection between a parametric line and such a surface (by plugging the line’s equation into the surface)

Curves

Parameterizations

Know the definition of parametric curves in both 2 and 3 dimensions. Understand how to tell which direction a parametric curve is going (drawing ‘arrows along the curve’).
Know how to parameterize a line given a point and direction, or given two points it passes through.
Understand that there may be multiple different parameterizations for the same curve, and be able to tell when this happens (if the coordinates of each parameterization satisfy the same relationship: like $(t,t)$ and $(t^3,t^3)$ both satisfy $y=x$).
Know the parameterization of a circle: be able to parameterize circles going clockwise or counterclockwise, of any radius centered at any point in the plane.
Be able to parameterize the intersection of surfaces in space, by using the equations of the surfaces one at at time to solve for $x,y,z$ in terms of a parameter $t$. (Example: intersection of a cylinder and a plane)
Understand how operations like scalar multiplication affect the shape of a parametric curve. Be able to turn a circle into a spiral or a helix, etc by changing the parameterization slightly.

Calculus

Know how to take limits of a parametric curve (via the limit of each component)
Know why you can take the derivative of a parametric curve component by component (applying the limit definition of the derivative, and then breaking it up into individual limits using the first point.)
Understand the geometric meaning of the derivative (a tangent vector to the curve) as well as its physical meaning (the velocity the particle tracing out the curve is moving), and of its magnitude (the speed).
Find the unit tangent vector to a curve and tangent lines to curves.
Know how to differentiate vector functions with different products (we have three products now, scalar multiplication, the dot product, and the cross product - so we have three product rules)!
Be able to integrate a vector function, and understand its meaning: recovering velocity from acceleration, or displacement from velocity.