The intro­duc­tory courses for math majors are MAT 215: Sin­gle Vari­able Analy­sis, MAT 217: Lin­ear Alge­bra, and MAT 218: Mul­ti­vari­able Analy­sis. Like the great major­ity of math courses at Prince­ton, these three courses are the­o­ret­i­cal and proof-based. The intro courses, espe­cially MAT 215, empha­size this mind­set and are geared towards teach­ing stu­dents to read and write proofs.

These three are usu­ally the first math classes that math majors take at Prince­ton. How­ever, the math depart­ment is very flex­i­ble in allow­ing advanced fresh­men to skip some or all of these courses. Stu­dents who do skip any of these three should make sure they are very com­fort­able with the cor­re­spond­ing mate­r­ial. Another note is that rarely, peo­ple opt to take MAT 214: Num­bers, Equa­tions, and Proofs instead of MAT 215. The for­mer is a course which also teaches proofs and rig­or­ous think­ing, but in the con­text of clas­si­cal num­ber theory.

Brief Course Sum­maries [Show]Brief Course Sum­maries [Hide]
The goal of MAT 215 is to build the the­ory of analy­sis from the ground up, teach­ing stu­dents to think rig­or­ously along the way. The course starts by address­ing the ques­tion: what are real num­bers? It then intro­duces its stu­dents to impor­tant topo­log­i­cal pre­lim­i­nar­ies such as open and closed sets, com­pact­ness, and com­plete­ness. The remain­der of the course is spent on devel­op­ing the the­ory of lim­its, dif­fer­en­ti­a­tion, inte­gra­tion, sequences, and series. See below for a first-hand descrip­tion of MAT 215.
MAT 217 is a course in lin­ear alge­bra, a sub­ject at the foun­da­tion of almost all branches of pure and applied math. The most basic math­e­mat­i­cal object this course deals with the vec­tor spaces, a struc­ture whose ele­ments can be added and mul­ti­plied by scalars. One exam­ple of this is the set of n-tuples of real num­bers. The major­ity of the course is spent study­ing lin­ear trans­for­ma­tions between vec­tor spaces and their close rel­a­tives, matri­ces.
MAT 218 is in a sense a con­tin­u­a­tion of MAT 215: it gen­er­al­izes the con­cepts of lim­its, dif­fer­en­ti­a­tion, and inte­gra­tion from one to mul­ti­ple dimen­sions. Though some of the mate­r­ial at the begin­ning of MAT 218 might look famil­iar, fairly soon analy­sis in sev­eral vari­ables takes on a fla­vor of its own. In par­tic­u­lar, lin­ear alge­bra turns out to play a sig­nif­i­cant role, espe­cially the space Rn and the deter­mi­nant. The course briefly touches on the sub­ject of man­i­folds, i.e., smooth sur­faces, which are impor­tant in fields such as topol­ogy, dif­fer­en­tial geom­e­try, and Lie the­ory. MAT 218 con­cludes with a sur­pris­ingly ele­gant gen­er­al­iza­tion of the fun­da­men­tal the­o­rem of cal­cu­lus called Stokes’ The­o­rem.

First-hand Account of MAT 215 [Show]
First-hand Account of MAT 215 [Hide]
For most stu­dents, MAT 215 will be the first course they take in rig­or­ous math­e­mat­ics. High school math typ­i­cally involves apply­ing and some­times slightly mod­i­fy­ing a pro­ce­dure that the teacher has pre­vi­ously demon­strated. In 215, you will never have a prob­lem this rou­tine. There are some recur­ring tech­niques that you will learn to apply, but home­work and exams will require you to maneu­ver a lit­tle dif­fer­ently in each instance. Lec­tures fol­low a com­mon pat­tern: the pro­fes­sor presents a the­o­rem, sup­plies a proof or sketch of a proof, and dis­cusses the sig­nif­i­cance of the result. Rarely will your teacher stop to address the best way to han­dle a par­tic­u­lar class of prob­lems. A few ques­tions will be more com­pu­ta­tional in nature (e.g. does the fol­low­ing series con­verge?), but even these will go far beyond basic approaches (e.g. apply­ing the root or ratio test, though both make an appear­ance in the course). If this seems fright­en­ing, relax. A cen­tral goal of the course is to build your famil­iar­ity with this style of math. The begin­ning of the semes­ter will be an adjust­ment, but even­tu­ally, you will become com­fort­able with these sorts of ques­tions.
For the stu­dents who choose to enroll in 215, it will likely be their most demand­ing and time-consuming course that semes­ter. Some will have enough pre­vi­ous expe­ri­ence to com­plete prob­lem sets in a few hours, such as those who have par­tic­i­pated exten­sively in math com­pe­ti­tions, but most should expect to work a min­i­mum of 10–15 hours on assign­ments; with high prob­a­bil­ity, there will be at least one prob­lem set that takes 20+ hours. Stu­dents should also spend some time each week review­ing class notes so that they can fol­low along with proofs dur­ing lec­ture– as with any course that builds upon ear­lier mate­r­ial, falling behind is a bad idea. Basi­cally, you should be ready to work really hard. But there is good news: 215 is famous for its large study groups. I rarely worked with fewer than 6 other peo­ple– all of us strug­gling, but strug­gling together. Office hours usu­ally turn into sem­i­nars, with the major­ity of the class attend­ing for home­work help. 215 is the clos­est that you will come to team math in your time at Prince­ton. As an incom­ing fresh­man, you will have a ready-made group of friends who will share your inter­est in math. These study groups became some of my favorite mem­o­ries from school.

Recent Changes [Show]Recent Changes [Hide]
The math depart­ment has recently seen its enroll­ment rise, which entails an increas­ing diver­sity in lev­els of prepa­ra­tion. Some incom­ing math majors will have already seen and done proofs, while oth­ers will have not. To accom­mo­date both of these groups, the math depart­ment will be split­ting up MAT 215 into two sec­tions. One sec­tion will intro­duce stu­dents to proofs more fully and grad­u­ally, while the other will assume expe­ri­ence with proofs and launch right into the mate­r­ial. Logis­ti­cally, the math depart­ment plans to split the two sec­tions like PHY 103 and PHY 105: have the two sec­tions start out together, but then use the first one or two prob­lem sets to dif­fer­en­ti­ate between the two groups.
In addi­tion, many math majors would like to have their intro­duc­tory math courses done by the end of fresh­man year. This has resulted in the low pop­u­lar­ity of the last intro course, MAT 218. In response to this, the depart­ment is now con­sid­er­ing switch­ing to a two-semester intro­duc­tory sequence. Lit­tle is know at this point about how or when this will happen.

Links to the Math Depart­ment Web­site [Show]Links to the Math Depart­ment Web­site [Hide]
The above descrip­tions for the intro­duc­tory courses are fairly short, due to the fact that the math depart­ment has done a good job of pro­vid­ing infor­ma­tion about these courses on its web­site. Please see here for more details on course con­tent, sam­ple mate­r­ial, and other infor­ma­tion. The FAQ sec­tions of the pages for the intro courses pro­vide answers to ques­tions such as “how hard is this course” and “is this the right course for me”.

Con­tacts [Show]Con­tacts [Hide]
Chris McConnell ’15 (cmcconne[at]princeton[dot]edu)
Gene Kat­se­vich ’14 (ekatsevi[at]princeton[dot]edu)