Good luck on the final today!
12/10
12/8
Today we worked in groups on review problems for the exam. I strongly suggest working on problems out of the book as well, especially the chapter 16 review. Otherwise make sure you study and prepare for the final on Monday, good luck everyone!
12/5
Today we went over all the theorems we talked about in class so far. This included Green's theorem, Stoke's theorem, the Divergence theorem, and others. We noticed how all of our theorems somehow involve integrating the derivative of something on one side, and integrating something over the boundary on the other. This is because all of these theorems are a special case of something called the generalized Stoke's theorem.
12/3
We finished discussing the Divergence theorem and did several more examples. We also talked briefly about how we can use the Divergence theorem to give an interpretation of the divergence of a vector field.
11/30
Today we introduced the Divergence Theorem. This is another generalization of Green's theorem, similar to Stoke's theorem. We did a few examples, and we will continue with examples on Monday. Also, please fill out the course evaluation if you have not yet.
11/28
We continued talking about Stoke's Theorem. We used it to compute line integrals and surface integrals. We also saw how the surface integral of the curl of a vector field only depends on the boundary. So if two surfaces share the same boundary, any surface integral of the curl of a vector field will be the same.
11/26
Today we went over the exam and introduced Stoke's Theorem. Questions about any grading on the exam should be sent to me or the TA by next Monday. Please follow the same protocol as before.
11/21
Good luck on the exam today!
11/26
Welcome back from break. Today we went over the midterm and we sketched (roughly) the solutions. We also stated Stoke's Theorem and will go over examples next time. Finally, requests for regrades will be the same as for the first midterm. Please hand these to me before next Monday.
11/19
Today we reviewed for the midterm by going over the midterm review and any other questions people had. Make sure you study hard for the exam on Friday!
11/16
We finished discussing surface integrals, and then defined what it means to integrate a vector field over a surface. This was similar to the line integrals of vector fields we looked at before. Remember that we need the surface to be oriented in order for this to make sense.
11/14
Today we introduced surface integrals. This is essentially the same as when we computed surface areas only we multiply by a scalar function in the integrand. We also looked at graph surfaces again. We will continue talking about surface integrals on Friday.
11/9
We finished talking about tangent planes, parametric surfaces, and calculating surface areas using parametrizations.
11/7
Today we introduced parametric surfaces. We did several examples of sketching paramteric surfaces. We also found a parametrization for several types of surfaces such as planes determined by vectors and revolutions of graphs. Next time we will discuss tangent planes.
11/5
We quickly reviewed Green's theorem, curl, and div. Then we introduced the Laplace operator and went over a few examples.
11/1
Today you introduced curl and divergence with the sub.
10/31
Today you covered Green's theorem with the sub.
10/29
Today we went over the midterm solutions, and then introduced Green's theorem. We only did one small example, but we will spend a lot more time on Green's theorem on Wednesday. If you have a regrade request, please write a short paragraph on a separate piece of paper explaining why you were correct. Then hand in that paper along with your exam to the sub or Jianghao by Friday's lecture.
10/26
Good luck on the exam today everyone!
10/24
Today we reviewed for the midterm on Friday. We spent the class period working on the midterm review that was emailed out earlier. If you did not get a copy of this review please let me know. Also, we found that there were a few typos in the review, so please be aware of this. I will try to get a more polished version handed out shortly. Also, the front page of the review will be the same as the front page of the actual midterm. This means that the rules will be the same: no calculator, and one handwritten sheet (two sided) of notes are allowed for the exam.
10/22
After reviewing some notation for line integrals over vector fields, we stated the fundamental theorem of line integrals. We also compared the theorem to the fundamental theorem of calculus. After that, we introduced several definitions such as closed curves, simply-connected domains, and stated a theorem that tells us exactly when a vector field is conservative over a simply-connected domain.
10/19
Today there was a substitute for lecture. You should have covered line integrals with respect to x and y. This is used to define line integrals through vector fields. The relation between line integrals and conservative vector fields will come up shortly.
10/17
Today we talked more about conservative vector fields, and saw one necessary condition for a vector field to be conservative. We also introduced line integrals, which are based on the arc length formula you should have seen in one-variable calculus. These integrals are with respect to arc length. On Friday we will discuss line integrals of vector fields.
10/12
Today we started talking about general changes of coordinates. This is essentially a higher dimensional analoug of u-substitution from single variable calculus. We introduced the jacobian of a transformation of the plane. Next time we will talk about what happens in three dimensions. We also saw how polar coordinates is just an example of a particular transformation.
10/10
Today we finished talking about spherical coordinates. We started with the examples of regions from the last lecture and then spent a while on some examples of triple integrals in spherical coordinates. Make sure you try a few more integrals in the homework or the book for more practice.
10/8
We talked more about spherical coordinates. We went over the surfaces left from the previous lecture and talked about triple integrals. For integration, we worked out what the change in differentials is and did some practice examples. We ended class by trying to describe some regions in 3-space using spherical coordinates.
10/5
Today we did an example of a triple integral in cylindrical coordinates. We talked about how to write the projection in the xy-plane in polar coordinates and then find the range of the z-values. At the end of class we started discussion spherical coordinates. We came up with the formulas and then I left an excercise of sketching some surfaces.
10/3
We introduced cylindrical coordinates. These are very similar to polar coordinates, only they include an extra "z" coordinate. As such, triple integrals in cylindrical coordinates require an extra "r" in the different when converting from cartesian. We talked a bit about where the "r" comes from in polar coordinates.
10/1
Today we reviewed triple integrals. We did not get through too many examples, so if you still feel uncomfortable with them please review more in the textbook. There are also some homework problems dedicated to triple integrals that should help.
9/28
Welcome to Math 2E! In this course we will learn about calculus of several variables. Topics will include triple integrals, line integrals, curl, div, Green's theorem, and more. Today we reviewed double integrals and polar coordinates.