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.
I will not test you on: The vector triple product \(a\cdot (b\times c)\) - even though you do already know the dot product and the cross product!
Shapes
- Know both the implicit equations and parametric equations for a line. Be able to find the equation of a line given two points, or a point on the line and a direction vector. Be able to find the normal vector to an implicit line.
- Know both the implicit equations and parametric equations for a plane.
- Be able to find the normal vector to an implicit plane and to a parametric plane.
- Be able to find the equation of a plane (whichever form is easier for you) given some information (for example: a point and a normal vector, or a point and two vectors, or three points)
- Know the implicit and parametric equations for a circle of radius \(R\) centered at \((h,k)\).
- Know the implicit equation for spheres in \(\RR^3\), and be able to find their center and radius (perhaps by completing the square)
- Understand the relationship between the equation for a circle in 2D and a cylinder in 3D.
I will not test you on: The parametric equation for a sphere, or the other shapes we briefly talked about (hyperboloids, ellipsoids, etc.)
Curves
Parameterizations
- Know the definition of parametric curves in both 2 and 3 dimensions. Be able to parameterize graphs of functions.
- 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 \(y=x\)).
- 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, from class)
- 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 hte 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).
- 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.
Geometry
- Be able to write down the arc length integral for a curve. Be able to compute the actual arclength when this integral is simple.
- Determine whether or not a curve is parameterized by arclength by checking if its derivative is unit length.
- Find the unit tangent vector to a curve.
- Know the definition of the normal vector to a curve, and be able to find the normal vector in simple cases.
- Know the formula for the curvature of a parametric curve \(\vec{r}\) in terms of its first and second derivatives. Understand how we were able to simplify this when the curve was a graph \(r(t)=(t,f(t),0)\).
- Understand the relationship between curvature and the radius of the best approximating circle.
Differential Equations
- Know what a vector valued differential equation is, and how to interpret it as a system of differential equations.
- Understand how to derive that objects fall on parabolas under a constant downward gravitational acceleration. Be able to solve simple differential equations like this (where the individual functions do not depend on one another.)
- Be able to take an almost solution and adjust it to make it an actual solution by changing its speed (like we did for circular orbits in Newtonian gravity)
Because this topic was our most recent topic (and thus there is no webAssign questions on it in our previous homeworks), I wanted to give a couple example questions below, that could be reasonalbe to ask.
- If \(c=(x(t),y(t))\), find a solution to \(c^\prime = \langle 2t, e^t-1\rangle\)
- Show that \(c(t)=(\cos(t),\sin(t))\) solves the differential equation \(c^\prime(t)\cdot c^\prime(t)=1\).
- The curve \((x(t),y(t))=(t,t^2)\) is almost a solution to the equation \(x y^\prime = y\). Can you adjust one of the component functions by a constant to make it a solution?