Mondays 18:00-19:00
Wednesdays 14:00-16:00
MWF
13:00 - 14:00
We will be using Walter Rudin's Principles of Mathematical Analysis.
All material relevant to the course will be covered in lecture, and assignments will be posted on the course website. The course textbook is an optional resource.
In past, some offerings of Math 104 have used Kenneth Ross's Elementary Analysis. Much of the content is the same as we will be covering, although presented in a different style and order; those wishing another perspective (or more practice problems) may find it useful to consult. It can be downloaded for free while on campus here.
The official course description can be found here.
We will begin with an examination of the real number system $\R$: codifying several properties you are already familiar with and discussing the least upper bound property. This latter property, which is crucial to the study of analysis, essentially tells us that $\R$ is solid enough for us to take limits, derivatives, etc. Also, we will take the real numbers $\R$ as given rather than constructing it from scratch (e.g. via Dedekind cuts). This shortcut will allow us to spend more time considering more general objects known as metric spaces.
Metric spaces are sets with a notion of "distance" on them. The real numbers are an example of such a space, where we measure the distance between two points $x,y \in \R$ by $\abs{x - y}$. The Euclidean plane $\R^2$ gives another example, where the distance between two points $(x_1, y_1), (x_2, y_2) \in \R^2$ is given (unsurprisingly) by the distance formula: $$\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$$ But metric spaces can be considerably more general: for example, it turns out that real-valued continuous functions (another of our core topics) have a suitable notion of distance and can therefore also be studied as a metric space. Metric spaces are of independent mathematical interest and provide nice examples and intuition for the theory of general topology.
To study differentiation, we will return to the more concrete context of real-valued functions on the real numbers $\R$. We will give a formal definition of the derivative and prove the theorems you used frequently in calculus: the product rule, the quotient rule, the chain rule, the mean value theorem, and Taylor's theorem. We will also analyze the Riemann integral and discuss what it means for a function to be integrable. Our examination of derivatives and integrals will culminate with the Fundamental Theorem of Calculus.
The course will conclude with a study of infinite series and power series, which are again concepts you have seen in calculus but which will be treated more formally here.
My aim is to foster an open and inclusive atmosphere in class. Therefore questions, participation, collaboration, and curiosity are strongly encouraged. Math can be hard, especially when we aren't honest with ourselves about whether or not we understand something. Confusion is not a sign of weakness, nor is asking for help. If you need help beyond class time and office hours, please do not hesitate to contact me so that we can work out additional times to meet.
Homework assignments will be posted on the course webpage, and will be collected via Gradescope. There will be an assignment due on Friday most weeks of the course, with some exceptions due to holidays and the midterm. No late homework will be accepted. Your lowest 2 homework scores will be automatically dropped.
Collaboration is strongly encouraged; you should be working with your peers to understand and solve the homework problems. However, your written proofs must clearly be your own, and indicate that you understand the argument.
On each homework assignment, there will be a 10% bonus available for submitting typed solutions.
The midterm and final exam will be administered similarly. The exams will be posted on the course bCourses website, and submission will be open via Gradescope for a period of 24 hours from when the exam is posted. You must solve the problems on your own, without assistance from others, without searching online, and without posting the questions to any website. You may consult your notes from the term and the course text book.
During the exams, you are expected to follow the Berkeley Honor Code:
The registrar has assigned this course an exam slot on Friday Dec. 16th from 19:00-22:00. The exam will be available for a 24 hour period including these three hours, and it is intended to take no more than 3 hours to complete.
There are two exam-based grading schemes offered, and one presentation-based scheme. If you write the final exam, I will automatically select the one which gives you the better grade. They schemes are as follows:
| Homework | Midterm | Final exam | Final presentation | |
|---|---|---|---|---|
| Scheme 1: | 30% | 30% | 40% | 0% |
| Scheme 2: | 30% | 15% | 55% | 0% |
| Scheme talk: | 30% | 30% | 0% | 40% |
The assignments and examinations in this course will be challenging. Your grade will not be curved based on the performance of the class as a whole, but will take into account the difficulty of the course. In the previous offering of this course, a grade of 80% was sufficient for an A+.
The bCourses site will be used to maintain a gradebook for the course. If you believe there is an error with the grading of any course material, you must notify the instructor within 14 calendar days of when it was completed; otherwise your complaint will not be given further consideration. The general course policies of the University of California, Berkeley may be found here.
Optionally, you may give a final presentation in lieu of writing the final exam. More information, should you choose to do so, is provided here.
UC Berkeley complies with the regulations of the Americans with Disabilities Act of 1990 and offers accommodations to students with disabilities. If you are in need of classroom or testing accommodations, please make an appointment with the DSP office to discuss the appropriate procedures. More information is available on their website.