Study Guide III
Write a study guide for yourself in preparation for the final exam. 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 solutions 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!
The final exam will focus mostly on Integration, starting with the theory of double & triple integrals, and proceeding through line integrals and circulation / flux of vector fields. A good resource to review the topics covered is (1) these course notes (2) past WebAssigns and Gradescopes, and (3) Chapters 15 and 16 in our textbook. Another great source of practice problems is Pauls Online Math Notes which has a huge collection of multivariable calculus problems and their solutions. To help you in formulating your study guide, I’ve additionally listed the main topics covered below.
Coordinate Cheat Sheet
I will give you the following formulas on the exam, as written below!
Cartesian Coordinates \((x,y)\) in 2 dimensions or \((x,y,z)\) in 3 dimensions. Area element for two dimensions is \(dA = dxdy\). Volume element for 3D is \(dV = dx dydz\).
Polar Coordinates \((r,\theta)\) where \(r\) is the distance to the origin, and \(\theta\) is the angle from the positive \(x\) axis (measured counterclockwise). Conversion to cartesian: \(x=r\cos\theta\) and \(y=r \sin\theta\). Area element is \(dA = rdrd\theta\).
Cylindrical Coordinates Is choosing to use polar coordinates for two of the three coordinates in a problem. This could mean \((x,y,z)\) becomes \((r,\theta, z)\) if \(x,y\) are converted, or \((r,\theta,y)\) if \(x,z\) are converted. The volume element is the polar area times the remaining variable, so \(rdrd\theta dz\) or \(rdrd\theta dy\) in these two examples, respectively.
Spherical Coordinates \((\rho,\theta,\phi)\) are three dimensional coordinates recording the geometry of a sphere. \(\rho\) measures the distance to the origin, \(\phi\) measures the angle from the north pole (positive \(z\) axis) and \(\theta\) is the same as polar coordinates. These relate to cartesian coordinates by \(x=\rho \cos\theta\sin\phi\), \(y=\rho\sin\theta\sin\phi\) and \(z=\rho\cos\phi\). The volume element is \(dV =\rho^2\sin\phi d\rho d\phi d\theta\).
Multivariable Integrals
Double Integrals
- Be able to evaluate double integrals over a rectangle domain as iterated integrals
- Be able to find the bounds for double integrals over general domains by slicing (either \(dx\) first or \(dy\) first)
- Practice slicing a domain both ways and seeing which integral is easier to evaluate
- Use double integrals to find the area of a region by integrating \(dA\).
Triple Integrals
- Be able to apply the same skills learned for double integrals to triple integrals
- Be able to choose a useful order of integration from a description of the bounds
- Use triple integrals to find the volume of a region by integrating \(dV\).
Coordinate Systems
- Be able to convert a double integral with circular symmetry to polar coordinates to evaluate.
- Use polar coordinates in 3 dimensions to convert two variables to polar and leave the other alone (that is, cylindrical coordinates) to simplify some triple integrals.
- Use spherical coordinates to simplify triple integrals whose domains contain segments of a sphere.
Line Integrals and Surface Integrals
- Be able to integrate a scalar function \(f(x,y)\) or \(f(x,y,z)\) along a curve \(C\) with respect to arclength \(\int_C fds\).
- Be able to integrate a scalar function on a surface in three dimensions \(\iint_S f\, dS\) where \(dS\) is infinitesimal surface area. (For this we looked at the special cases that the surface was (1) planar so \(dS = dxdy\), (2) a sphere or cylinder, where we used spherical or cylindrical coordinates, or (3) the surface of a graph \(z=g(x,y)\) in space.)
Vector Fields
Derivatives
- Be able to compute the divergence of a vector field
- Be able to compute the curl of a vector field in 2D to get a scalar
- Be able to compute the curl of a vector field in 3D to get another vector field
- Understand how to interpret the curl or divergence of a vector field at a point.
Circulation (Work) and Flux Integrals
- Understand the derivation of circulation as the tangential component of a vector field along a curve, and of flux as the normal component of a vector field along a curve/surface.
- Compute the circulation of a vector field \(\vec{F}\) along a curve \(C\) by integratint \(\int_C F(c(t))\cdot c^\prime(t) dt\).
- Compute the flux through a curve in 2D by integrating \(\int_C F(c(t))\times c^\prime (t)dt\)
- Compute the flux through a surface in 3D by the surface integral \(\iint_S \vec{F}\cdot N dS\) where \(N\) is the normal vector to the surface.
Fundamental Theorems
- Be able to find a potential function \(f\) for a vector field \(F\).
- Use the curl to tell when no potential \(f\) exists.
- When \(F\) has a potential, be able to compute the line integral using the fundamental theorem for the gradient.
- Compute line integrals around closed curves by using the fundamental theorem for curl (Stokes’ Theorem)
- Compute flux integrals around a curve / surface by using the fundamental theorem for divergence (Gauss’ Theorem)