Is there Life after Calculus?

You've reached the end of your first year, and are probably thinking about your second. Whether math is your thing or not, there's probably more in your future, and this page might help.

Which describes you?

I like Math, and I'm thinking of trying some more math courses...

Math is ok I guess, but why would I want to take more?

 No, I don't much like math, but my program requires Math YYY...

I want a degree in Biology/Physics/Chemistry/CIS/Etc... What math courses would be useful/marketable?

I'm not crazy about  theory, but I really like calculating things, and working with numbers...

I'm wondering about doing a degree in mathematics or statistics....

I'm thinking about teaching high school...

What about mathematics after calculus? What is there?

Don't you have to be a genius to study math?

Can you get a job with a math degree? Where are the jobs?

I have to take Math XXX for my program. What's that about?

I like Math, and I'm thinking of trying some more math courses...

Great! UFV has some second-year courses which will give you a taste of what lies beyond calculus. There's a field guide to what you might find interesting and useful below.

Field guide to second-year mathematics courses:
  • Math 211: Calculus III Calculus goes 3D! The central ideas of calculus are expanded to work with functions of more than one variable. This is a core course,  and a gateway to many upper level courses. Get this right away. Read more here... Offered in Fall, each year.
  • Math 255 Ordinary Differential Equations  When calculus techniques meet the real world, it's very frequently via a differential equation. "DEs" describe population change, fluid flow, knowledge acquisition, you name it. Read more here... Offered in Fall, each year.
  • Math 270 Introduction to Statistics  Whether it's politics (opinion polling) industry (designing a quality test) science (extracting relationships from data)  there are not many parts of the modern world that don't employ data and its analysis. What are the tools needed?  Read more here...�
  • Math 221 Linear Algebra It's been said that you can never know too much calculus and you can never know too much linear algebra. A course full of brand-new ideas and techniques that get applied all over mathematics, science and technology. The subject that provides some of the mathematical ideas behind medical imaging (CAT scans), internet searching (Google calculates "eigenvalues" of an order 25 billion square matrix), and quantum mechanics. Its concept and structures are fundamental to much of mathematics. Absolutely fundamental, and a gateway to many upper-level courses. Get this in place as soon as you can. Read more here... Offered in Winter semester, every year.
  • Math 265 Transition to Advanced Mathematics This is a course for those who enjoy mathematics, and want to learn more. It's not a course about calculation, it's about ideas. On the one hand, you'll learn tools to construct proofs and arguments, and on the other learn some amazing new things. (It turns out, for example, that one can make perfectly good mathematical sense of the idea of infinity, and that there's different levels of it! Infinitely many different levels...) It is here as well that you start to put in place some of the theory behind calculus. This course is essential for those wanting to go on to upper-level mathematics courses, and a degree requirement. Offered in Winter semester, every year. 

Don't know what to choose? Try Math 211 and 270 in the Fall, and go from there. We'd be happy to talk to you: email us here

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Math is ok I guess, but why would I want to take more?

Math may not be your first love, but whatever area of science or technology catches your fancy, mathematics and statistics will be there. Some things to think about as you plan:

  • Adding even a few extra math courses to your transcript can
  • signal to prospective employers that you have the ability to think analytically (ie you can work things out), abstractly (you can think beyond what you see in front of you) and logically.
  • help improve your performance in your other science courses
  • One of the most effective (and attractive) degree combinations is a major in a science (like biology, or computer science say) and a minor in mathematics. Having that minor can make you the "go-to" person in the workplace who has a good understanding of the issues underlying the data, who can read and interpret the literature, who can design the more efficient procedure, or just think more clearly and creatively about the issues at hand. More about making yourself marketable via a math minor.
  • You probably like math most when it's "applied", when it's clear how it can be used in the real world. Here are some applied courses that would be an excellent supplement to any degree you decide to do:
    • With only an introductory statistics course (like Math 270) there are several upper-level applied statistics courses you can take, like Math 315 (Applied Regression Analysis), 330 (Design of Experiments) or 350 (Survey Sampling). Why study statistics? Every area of science and technology deals with data, whether designing ways to gather it, analyzing it, or using it to make predictions. That's what statistics is about. If you're the person in the workplace who understands the principles and methods, you're a valuable person!  Read more here
    • Second-year math courses can take you lots of useful places:
    • Math 211 (Calculus III) : A next step in calculus with many applications. The language and techniques used are are used in areas from business to biology. Read more here... Offered in the fall semester
    • Math 221 (Linear Algebra) or Math 152: The tools provided here (matrix methods) are essential computational devices applied everywhere from computer science to population biology. Math 152 is a more applied course, but Math 221 keeps more doors open at the upper levels. Read more here...  Offered in Winter semesters
    • Math 255 (Ordinary Differential Equations): When calculus techniques meet the real world, it's very frequently via a differential equation. "DEs" describe population change, fluid flow, knowledge acquisition, you name it. Read more here...  Offered in Fall semesters
  • There are many applied mathematics and statistics courses available at the upper level. An advisor can help you choose.

 
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No, I don't much like math, but my program requires Math YYY...

That's ok, just make sure you don't close any doors. A few comments about various programs:

Biology You'll need Math 104, Math 106 or 270 for your statistics requirement. The honours degree also requires Math 271.
Chemistry  You'll need at least one statistics course (Math 106 or Math 270 are your current options) as well as Math 211. Since Math 211 builds on your first-year calculus, it's a good idea to get this sooner, rather than later when you may have gotten rusty!
Physics There's a lot of math in your future, in courses from both departments. Course offering are sometimes limited in a small university like UFV. Getting your second year courses in place soon will increase your options in the following years.
The Physics major degree requires Math 211. Taking Math 255 (ordinary differential equations) will prepare you well for Physics 381, another requirement.
Other programs  It's always a good idea to talk to an advisor. If you're wondering about a specific math course, be sure to ask.


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I want a degree in Biology/Physics/Chemistry/CIS/Etc... What math courses would be useful/marketable?

 

Biology:

You're required to take a statistics course (Math 104 or 106 ) for the biology major, and with either of those in place, there are some very useful applied statistics courses available at the upper level. With courses like these you can be the "go-to" person in your workplace, with the analytical skils needed to help design the studies, understand the literature, or interpret the data. And they're easy to get into! Have a look:

 Make yourself valuable:  Add a math/stats minor (why?)

Physics: You can't really avoid taking a fair bit of math in a physics degree. But did you know that by the time you've met the requirements for a major in Physics you'll almost have a minor in mathematics? A Physics major requires 30 upper level credits, but the BSc itself requires 46 upper-level credits. So taking those extra 16 credits in upper-level math (with the right second-year courses in place) will get you a math minor!

You'll need Math 211 for your Physics degree (it's a requirement). Beyond that Math 255 (Differential equations) will give you a basic background in one of the fundamental tools Physics uses to model the real world. It's a good preparation for Physics 381 as well, also a Physics degree requirement.

Modern Physics subjects like quantum mechanics and relativity use the language of linear algebra; that's Math 221.

Make yourself valuable:  Add a math/stats minor (why?)

Chemistry  You're required to take a statistics course, Math 106 or 270. For someone with a calculus background Math 270 would be the best choice.  With either of those in place, there are some very useful applied statistics courses available at the upper level. With courses like these you can be the "go-to" person in your workplace, with the analytical skils needed to help design the studies, understand the literature, or interpret the data. And they're easy to get into! Have a look:

 Have a look:

Make yourself valuable:  Add a math/stats minor (why?)

CIS/Computer Science You'll need Math 125 for the CIS or Computer Science degrees. Get this soon if you don't already have it! With that in place you'll find the following second-year courses useful (with many more at the upper levels):

Math 225 Topics in Discrete Mathematics: This course introducess you to some of the most sueful types of combinatorial structures: graphs, trees, generating functions, and recurrence relations; all these play an important role in the mathematics of computers and computation.

Math 265 Transition to Advanced Mathematics: Here you learn the language of mathematics through careful statements of definitions and construction of proofs. Topics include strategies for writing proofs of theorems, and how to effectively communicate mathematics to others.

Math 221 Linear Algebra It's been said that you can never know too much calculus and you can never know too much linear algebra. A course full of brand-new ideas and techniques that get applied all over mathematics, science and technology. The subject that provides some of the mathematical ideas behind medical imaging (CAT scans), internet searching (Google calculates "eigenvalues" of an order 25 billion square matrix), and quantum mechanics. Its concept and structures are fundamental to much of mathematics. Absolutely fundamental, and a gateway to many upper-level courses. Get this in place as soon as you can. Read more here... 



Make yourself valuable: Did you know that there a mathematics minor available within the CIS degree? Read more here.

Other areas The sort of math or statistics courses useful to you varies a bit with what you decide to do. We can help you choose, or you might talk to an advisor from your "home" department.

Make yourself valuable:  Add a math minor (why?)

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I'm not crazy about  theory, but I really like calculating things, and working with numbers...Have you thought about statistics? Every area of science and technology deals with data, whether designing ways to gather it, analyzing it, or using it to make predictions. That's what statistics is about. If you're the person in the workplace who understands the principles and methods, you're a valuable person! For Today's Graduate, Just One Word: Statistics

The job market for people with a statistics background (even just a minor) and some programming experience is quite good. Pharmaceutical companies in particular are looking for people with SAS (Statistical Analysis Software) programming skills. 

You can build on you expertise in whatever field you study with UFV's brand-new Certificate in Data Analysis.


Here you can find a good description of what statistics is about, and possible careers.

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I'm wondering about doing a degree in mathematics or statistics....

Great! There are several math degrees available at UFV; you can find them listed here. That may seem like a long and confusing list: talking to a science�advisor can help you get things sorted out, and we have people in the math department who would be delighted to talk to you! Make an appointment here.

Do you wonder what mathematics looks like as you get beyond calculus? (Mathematical) life after calculus... 

What should you take next year? The requirements of the various degrees vary a little, but certainly Math 211 (Calculus III) and  Math 221 (Linear Algebra). Depending on your degree Math 265 (Transition to Advanced Mathematics) and Math 270 (Statistics) are also important. Don't wait to take these courses, as upper level courses have these as prerequisites! If you need help deciding, or you have to choose because your schedule is too full, we can help (just ask).
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I'm thinking about teaching high school...

You'll be much more in demand if math is one of your "teachable subjects." There's a chronic shortage of math teachers in BC (and elsewhere). Have a look at these statistics from SFU's teacher education program (PDP). With a math minor (or major) under your belt you're almost guaranteed a seat!

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Don't you have to be a genius to study math?

No. And the UFV math department faculty members are  proof.  (;  Kidding aside, all you need some interest, and some committment. Innate ability plays some role, but not as much as you might think. If you've made it to the end of first-year calculus you already understand more mathematics than 95% of the population, and that in itself speaks to your abilities.

The media portrays people who study math as either brilliant, or eccentric. But the truth is they're just ordinary folks who enjoy mathematics, like maybe you. Here are some career profiles of recent mathematics graduates

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Can you get a job with a math or statistics degree? Where are the jobs?
Yes!  Have a look at our careers page, and note in particular the healthy job prospects in statistics.

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I have to take Math XXX for my program. What's that about, anyway?

There's always the UFV calendar of course. But that doesn't tell you much. Have a look at our "friendly descriptions" page.

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(Mathematical) Life after Calculus...

The following is meant to give you a "big picture" view of some areas of modern mathematics. You'll get more specific information and suggestions about UFV courses here. This section is adapted from a Cornell university page.

Have you seen the best that mathematics has to offer? Or, as the title asks, is there (mathematical) life after calculus? In fact, mathematics is a vibrant, exciting field of tremendous variety and depth, for which calculus is only the bare beginning. What follows is a brief overview of the modern mathematical landscape:

Analysis

Analysis is the branch of mathematics most closely related to calculus and the problems that calculus attempts to solve. It consists of the traditional calculus topics of differentiation, differential equations and integration, together with far-reaching, powerful extensions of these that play a major role in applications to physics and engineering. It also provides a solid theoretical platform on which applied methods can be built. Analysis has two distinct but interactive branches according to the types of functions that are studied: namely, real analysis, which focuses on functions whose domains consist of real numbers, and complex analysis, which deals with functions of a complex variable. This seems like a small distinction, but it turns out to have enormous implications for the theory and results in two very different kinds of subjects. Both have important applications.

The study of differential equations is of central interest in analysis. They describe real-world phenomena ranging from description of planetary orbits to electromagnetic force fields, such as, say, those used in CAT scans. Such equations are traditionally classified either as ordinary differential equations (if they involve functions of one variable) or partial differential equations (if they involve functions of more than one variable). Each of these two corresponds to an active subfield of analysis, which in turn is divided into areas that focus on applications and areas that focus on theoretical questions.

Relevant courses at UFV: Math 265, Math 316, Math 322, Math 340, Math 415, Math 440, Math 444

Algebra

Algebra has its origins in the study of numbers, which began in all major civilizations with a practical, problem-set approach. In the West, this approach led to the development of powerful general methodologies. One such methodology, which originates with Euclid and his school, involves systematic proofs of number properties. A different methodology involves the theory of equations, introduced by Arab mathematicians ("algebra" itself has Arabic etymology). Modern algebra evolved by a fusion of these methodologies. The equation theory of the Arabs has been a powerful tool for symbolic manipulation, whereas the proof theory of the Greeks has provided a method (the axiomatic method) for isolating and codifying key aspects of algebraic systems that are then studied in their own right. A notable example of such fusion is the theory of groups, which can be thought of as a comprehensive analysis of the concept of symmetry. Group theory is an area of active research and is a fundamental tool in many branches of mathematics and physics.

The simplest and most widely known example of modern algebra is linear algebra, which analyzes systems of first-degree equations. Linear algebra appears in virtually every branch of applied mathematics, physics, mathematical economics, etc. Even though the theory of linear algebra is by now very well understood, there are still many interesting areas of research involving linear algebra and questions of computation.

If we pass to systems of equations that are of degree two or higher, then the mathematics is far more difficult and complex. This area of study is known as algebraic geometry. It interfaces in important ways with geometry as well as with the theory of numbers.

Finally, number theory, which started it all, is still a vibrant and challenging part of algebra, perhaps now more than ever with the recent ingenious solution of the renowned 300-year old Fermat Conjecture. Although number theory has been called the purest part of pure mathematics, in recent decades it has also played a practical, central role in applications to cryptography, computer security, and error-correcting codes. 

Relevant courses at UFV: Math 221, Math 308, Math 339, Math 355, Math 438, Math 439


Combinatorics (Discrete Mathematics)

Combinatorics is perhaps most simply defined as the science of counting. More elaborately, combinatorics deals with the numerical relationships and numerical patterns that inhere in complex systems. For a simple example, consider any polyhedral solid and count the numbers of edges, vertices, and faces. These are not random numbers; combinatorial analysis reveals their interrelationships. Practical applications of combinatorics abound from the design of experiments to the analysis of computer algorithms. Combinatorics is, arguably, the most difficult subject in mathematics, which some attribute to the fact that it deals with discrete phenomena as opposed to continuous phenomena, the latter being usually more regular and well behaved. Until recent decades, a large portion of the subject consisted of classes of difficult counting problems, together with ingenious solutions. However, this has since changed radically with the introduction and effective exploitation of important techniques and ideas from neighboring fields, such as algebra and topology, as well as the use by such fields of combinatorial methods and results.

Relevant courses at UFV: Math 125, 225, 360, 445

Geometry and Topology

These two branches of mathematics are often mentioned together because they both involve the study of properties of space. But whereas geometry focuses on properties of space that involve size, shape, and measurement, topology concerns itself with the less tangible properties of relative position and connectedness.

Nearly every high school student has had some contact with Euclidean geometry. This subject remained virtually unchanged for about 2000 years, during which time it was the jewel in the crown of mathematics, the archetype of logical exactitude and mathematical certainty.

And then in the seventeenth century things changed in a number of ways. Building on the centuries old computational methods devised by astronomers, astrologers, mariners, and mechanics in their practical pursuits, Descartes systematically introduced the theory of equations into the study of geometry. Newton and others studied properties of curves and surfaces described by equations using the new methods of calculus, just as students now do in current calculus courses. These methods and ideas led eventually to what we call today differential geometry, a basic tool of theoretical physics. For example, differential geometry was the key mathematical ingredient used by Einstein in his development of relativity theory.

Another development culminated in the nineteenth century in the dethroning of Euclidean geometry as the undisputed framework for studying space. Other geometries were also seen to be possible. This axiomatic study of non-Euclidean geometries meshes perfectly with differential geometry, since the latter allows non-Euclidean models for space. Currently there is no consensus as to what kind of geometry best describes the universe in which we live.

Finally, the eighteenth and nineteenth century saw the birth of topology (or, as it was then known, analysis situs), the so-called geometry of position. Topology studies geometric properties that remain invariant under continuous deformation. For example, no matter how a circle changes under a continuous deformation of the plane, points that are within its perimeter remain within the new curve, and points outside remain outside. For another example, no continuous deformation can change a sphere into a plane. So they are topologically distinct.

Topology can be seen as a natural accompaniment to the revolutionary changes in geometry already described. For, once one recognizes that there is more than one possible way of geometrizing the world, i.e., more than just the Euclidean way of measuring sizes and shapes, then it becomes important to inquire which properties of space are independent of such measurement. Topology, which finally came into its own in the twentieth century, is the foundational subject that provides answers to questions such as these. It is a basic tool for physicists and astronomers who are trying to understand the structure and evolution of the universe. Indeed, recent astronomical observations, together with basic results of topology, offer the exciting prospect that we will soon be in possession of the global topological structure of the cosmos.


Relevant courses at UFV: Math 340, Math 444

Probability and Statistics

Everyone has had some contact with the notion of probability, and everyone has seen innumerable references to statistics.

The science of probability was developed by European mathematicians of the eighteenth and nineteenth century in connection with games of chance. Given a game whose characteristics were known, they devised a way of assigning a number between 0 and 1 to each outcome so that if the game were played a large number of times, the number — known as the probability of the outcome — would give a good approximation to the relative frequency of occurrence of that outcome. From this simple beginning, probability theory has evolved into one of the fundamental tools for dealing with uncertainty and chance fluctuation in science, economics, finance, actuarial science, engineering, etc.

One way of thinking about statistics is that it stands probability theory on its head. That is, one is confronted with outcomes, say, of a game of chance, from which one must guess the basic rules of the game. So, statistics seeks to recover laws or rules from numerical data, whereas probability predicts (within some margin of error) what the data will be, given certain rules.

More information about statistics and its applications.

Mathematical Logic


Mathematical logic has ancient roots in the work of Aristotle and Leibniz and more modern origins in the early twentieth century work of David Hilbert, Bertrand Russell, Alfred North Whitehead, and Kurt Gödel on the logical foundations of mathematics. But it also plays a central role in modern computer science, for example in the design of computers, the study of computer languages, the analysis of artificial intelligence.

Mathematical logic studies the logical structure of mathematics, ranging from such local issues as the nature of mathematical proof and valid argumentation to such global issues as the structure of axiom-based mathematical theories and models for such theories. One key tool is the notion of a recursive function, pioneered by Gödel and intimately connected with notions of computability and the theory of complexity in computer science.

In addition to its contribution to mathematical foundations and to computer science, mathematical logic and its methods have also led to the solution of a number of important problems in other fields of mathematics such as number theory and analysis.

AND MORE...

This discussion has ignored many areas of modern "applied" mathematics.  


 Quick Links


What is statistics?

The world is becoming quantitative. More and more professions, from the everyday to the exotic, depend on data and numerical reasoning.

Data are not just numbers, but numbers that carry information about a specific setting and need to be interpreted in that setting. With the growth in the use of data comes a growing demand for the services of statisticians, who are experts in the following:

  • Producing trustworthy data
  • Analyzing data to make their meaning clear
  • Drawing reliable conclusions from data

(from www.amstat.org)

As someone with a background in statistics, you can help people see the "big picture" lying in their data; you have the expertise to help them understand.

The job market for people with a statistics background (even just a minor) and some programming experience is quite good. Pharmaceutical companies in particular are looking for people with SAS (Statistical Analysis Software) programming skills.


Here you can find a good description of what statistics is about, and possible careers.


Making yourself marketable: math or statistics minors

A particularly effective degree is one which combines expertise in some science (say Biology or CIS/Computer Science) with a mathematics or statistics minor. With that in hand, you have the advantage of a degree in a rapidly expanding field you enjoy (your major) along with the analytical skills and mathematical tools provided by your minor. Whether you end up in a lab, a business, or industry there will be mathematics and statistics behind the experiments done, the models used, or the computational routines employed. But in many situations companies do not have the in-house mathematical or statistical expertise to support and develop their enterprise. That extra edge can make you quite a valuable person!

Note that adding a minor need not increase the number of upper-level credits you need for a degree. Typically a B.Sc. major degree will require 44 upper-level credits, for example, but only 30 of them in your "major" area. The other 14 credits are only one credit short of enough for a minor. (You will need to increase the number of second-year courses you acquire though.)

Another option is to take our new 10-month Post-degree Certificate in Data Analysis after finishing your degree. The post-degree certificate builds on the skills and knowledge you have already acquired in earning your first degree allowing you to employ them fully in modern data-driven enterprise. The combination of your degree background and your data analysis skills can make you an attractive employee.


There are several math degrees available at UFV; you can find them listed here. That may seem like a long and confusing list: talking to a science advisor can help you get things sorted out, and we have people in the math department who would be delighted to talk to you! Make an appointment here.


What math courses will be offered next year?

See our upcoming courses page


Want a really good job?

One of the careers which is has been consistently rated as best or near-best in North America is that of an actuary, a job for which one needs quite a bit of mathematics. An actuary is an expert in evaluating the probability of future events, assessing risk, and designing ways to achieve the best possible outcomes. It's a challenging job with a good salary (about $90000 in 5-9 years), excellent opportunities, and the job market is strong.

You can read more about being an actuary here.

(In fact, according to the "Jobs Rated Almanac" a rating of jobs in North America taking into account salary, benefits, stress, working conditions etc.) six of the top ten jobs on the list need a good amount of mathematics training!)

 

 

 

 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  


 

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