# What kind of math do engineers really use? [closed]

I am from the mathematics StackExchange section, many of my students are engineering students at University. I was wondering what kind of calculus do you real engineers use? I have known two engineers. One from airplane design and another from metrology. The former used very very little calculus, some ODE's with constant coefficients by linearization. The latter used only basic math, no calculus, with some excel. I want to be honest with any engineering student so they know what awaits them.

Also, a follow up question. Did you find it beneficial to have about four semesters of calculus? Maybe you do not use anything from it, but it does enhance your mathematical reasoning, which does have a positive externality on your engineering skills?

• The follow up question might be a bit problematic for stackexchanges because it is kind of a poll which usually do not work well. Surely with four semesters one should get a thorough background knowledge. But then the question is what one could learn alternatively in the time. Very difficult to judge. Feb 9 '15 at 8:35
• I think we can have soft questions with subjective but definite answers, as long as they are not polls, somewhat like Workplace.SE. (1) I suggest the follow-up is a different question, so that your first question does not get caught up in a debate, or exclude engineers whose maths education was not divided into semesters. (2) It is less of a poll if you ask: "What benefits are there for practising engineers from learning a complete applied calculus course, including ODEs, PDEs, complex analysis ... ?". Then we can answer from published sources and experience of our colleagues work etc. Feb 9 '15 at 8:43
• I have commented on the follow up question on Meta: meta.engineering.stackexchange.com/questions/151/… Feb 9 '15 at 9:29
• A lot of engineering is about taking shortcuts to get results, so we use things like Fourier transform tables to avoid doing calculus: math.stackexchange.com/a/67461/2206 Feb 9 '15 at 14:46
• This depends on what you mean by "use". Compute an integral by hand? No, not really. But if I didn't know how to set up integrals I wouldn't understand the physics I've needed, and what sort of quantitative relationships are relevant to a problem. I use it everyday in the sense that someone who had never learned it wouldn't be able to do any of the engineering jobs I've had, even if they didn't ever explicitly need integration by parts or whatever. Feb 9 '15 at 19:08

In my civil engineering degree we used ODEs for the relationship between force, moment and deflection. I don't remember using PDEs myself, but my brother-in-law (doing civils at a different university) used them for hydraulics.

In real life (as a bridge designer) I can't remember actually using calculus. University mainly concentrated on the theory and the mathematical models used, whereas in actual engineering design we have computer software that does all the calculation for us.

I think there is a lot of benefit to a theoretical and mathematical background at university - as a professional engineer you need to have a basic understanding to know whether the software is giving you a sensible answer.

(As an aside, as you mentioned Excel, I've used that a hell of a lot in real design.)

• Thank you for the answer. I will let my students know that actual engineers do not use that much calculus, if not any at all. However, knowing some calculus and how some of it works, is very beneficial for solving an engineering problem. For example, maybe one has to change a computer program to fit some new model? Knowledge of some calculus may be helpful here. Feb 9 '15 at 9:09
• I graduated nearly 30 years ago. My experience is similar to AndyT. I have never used calculus, nor did anyone I worked with. I have used some trig, algebra & statistics + financial calcs for project evaluations, NPV, IRR etc. Much usage of computer design software & spreadsheets since uni. 2/3 to 3/4 of maths I studied in uni was never used. It was basically an exercise in how to think. Most useless maths unit for me was eigenvectors. Engineering courses need to be accredited by prof eng societies, hence there's a lot of maths, just in case its needed. Research engs use more maths & calculus
– Fred
Feb 9 '15 at 10:43
• @Fred your comment looks like a great answer. Feb 9 '15 at 11:06
• In terms of changing a computer program to fit a new theoretical model: most software used is proprietary software. The developers for that company might change it/add things to it, but the ordinary consulting engineer isn't going to have access to the source code to change/add anything. There are some people with engineering degrees who work in software but the vast majority don't do any programming. Feb 9 '15 at 12:52
• @dcorking - Fair enough. I should caveat my previous comment that it applies to my experience of civil engineers, rather than any other engineering discipline. Feb 9 '15 at 17:17

I originally wrote this as a comment attached to AndyT's answer but in response to dcorking's comment I've decided to expand here.

I graduated nearly 30 years and my experience is similar to AndyT's. After graduating I went straight into industry. Since graduating, I and everyone I have worked with or been associated with have never used and have never needed to use calculus in our day to day work as engineers. The types of engineers I have worked with include: civil, mechanical, ventilation, mining, electrical and environmental.

During my career I have used some trigonometry, algebra and statistics, plus financial maths (NPV, IRR, etc.) for project evaluations, feasibility studies and sometimes when I had to write or review justifications for capital expenditure.

When I emerged into the real world, work desktop computers were starting to be used by engineers. My early career was a mixture of doing designs on paper and using computers. Eventually computers dominated and I ended up using computer design software and spreadsheets for my engineering and design work.

Between two-thirds and three-quarters of all the maths I learnt in university I have never used after I started working. I have since realised that much of the maths I was required to learn was an exercise in teaching me how to think and solve problems. The maths unit that I particularly found useless for my career, but had to study, was eigenvectors. I know that some engineers find eigenvectors indispensable. It was one unit I was happy to forget after I sat the exam!

Engineering courses need to be accredited by professional engineering societies, hence engineers are required to learn a lot of maths, just in case it's needed. When students start their courses they don't always know where they will end up.

Research engineers and those involved with leading edge higher technologies use more of the maths and calculus they were taught.

I can recall overhearing a conversation one my lectures was having with another student and he said that the only time he used calculus was in the 1950s when he was involved in the design of certain types of internal combustion motors.

The thing about engineer's in industry is they soon end up being managers – looking after people, money and ideas. A background knowledge of calculus is useful but now days computers do all the complex calculations for us. We plug in the number and interpret the results. We need to know the concepts of how the software works to make sure the software is not giving us rubbish. It's one of the reasons why engineering students need to study maths.

I can recall attending a students meet industry seminar when I was a student and an experienced engineer told everyone that while at university they needed to use scientific calculators, but as they progressed through their careers they will end up using calculators that only had addition, subtraction, multiplication and division keys.

A little background (honest disclosure). I started out getting my B.S./M.S. in Mech Eng. from a fairly practical/applied school before deciding to continue with a PhD at a more theoretical school. As a result, I don't claim to be a real engineer (my general experience is that academics working in engineering are usually mediocre engineers), but I've got a few thoughts that might be helpful.

In my research, I find myself dealing with ODEs, PDEs, linear algebra (both applied and abstract) and that sort of thing. At times I've had to re-learn math concepts that I had forgotten or never learned in the first place. Whatever fraction of your students go into academia will be more likely to use calculus regularly.

In more applied activities, such as consulting projects or building race cars for a student completion. I find much less demand for those skills, although they are useful at times.

In many cases, calculus is more valuable for concepts than for actual computation. I'll want to know that one quantity is the integral of another in order to understand a problem, but that doesn't mean that I'm actually going to sit down and integrate an equation with a pencil and paper. Particularly, I think that understanding the basic ideas of differential equations can be extremely valuable across many disciplines (dynamic systems, heat transfer, electronics...).

The experiences that you describe are not unreasonable for quite a few reasons (not comprehensive list):

• Many practical problems may be solved by analytically with higher mathematics. However the analytic solution, once known reduces the actual computation to simple arithmetic. In some cases it's not only easier to use the given solution, but actually required. In the case of various codes and standards, an engineer would expose themselves to liability if they deviated from a prescribed computation procedure.

• Numerical solutions to problems are increasingly easy to come by and are more broadly applicable than analytic solutions. It's often easier to throw a numerical method at an integral, ODE, PDE, series... rather than try to remember/derive the solution. Complex geometry, non-linear behavior etc. often mean that conventional methods are impractical or impossible. And, with a lot of modern software, the math is totally invisible to the user. I've seen 1st year students with little experience quickly learn the tools to simulate stresses in complex load scenarios and compute transient heat conduction with non-linear boundary conditions (basically no math required).

• There's a whole lot of empirical data that goes into engineering. Experiments and experience may be just as good or better than mathematics in some cases. I couldn't even start to calculate (from first principles) the coefficient of friction between two materials, but I can look it up in a book or measure it myself.

• I upvoted your answer, but I would take issue with the implication that numerical and experimental methods are somehow not mathematical. For example, sometimes you need to be able to formulate your model as a differential equation, before you can use shrink-wrapped software to solve it. Feb 9 '15 at 8:37

This is from the view of a civil engineer.

Engineers typically don't use higher level math because the code specifications are written specifically to avoid the need. You don't want a building or bridge to fail because an engineer didn't take an integral correctly. Wherever possible, the hard math has been reduced to a simplified equation, a chart or a graph. This is done to limit the possible sources of errors.

The complicated math is done and checked before it is placed in the codes. This way the engineer that uses the code later does not have to worry about it being correct. Usually, just referencing a code is enough to "prove" that an answer is correct.

Engineering for the public is so controlled by codes and specifications that in some areas there is little math to actually be done. The answer is found in a table. The table was likely designed with lots of math input and university research, but a table was developed to eliminate the need to redo standard calculations on every project. This is even the case in seismic (earthquake) design. Unless a design is so special that a full computer model needs to be created, all of the complex interactions between the soil, the structure, and nearby faults is reduced to a simple horizontal load that is applied through the center of mass.

Building codes and uncertainties in loads require factors of safety to be somewhat large compared to other professions. This means that a simplified method for solving a problem doesn't affect the end result much when compared to an exact mathematical solution.

Much of the day to day calculations that an engineer completes use the same sets of formulae with different inputs. This is why huge Excel spreadsheets can be created to do a lot of the work.

This doesn't mean that higher level math and the theories that go behind it aren't useful though. All of those topics help to train an engineer's mind to visualize what is really going on. The topic on numerical simulation speaks to this.

• Aren't the codes written and checked by professional engineers who can do calculus? Feb 9 '15 at 14:34
• @dcorking: Yes, but a lot of the heavy research behind the codes is done at universities. That would stretch the limits of what would be called "typical" engineers. Also the ratio of engineers who use the codes to those that create them skews greatly to those using.
– hazzey
Feb 9 '15 at 14:41
• Your point about the ratio of civil engineers using codes, as opposed to developing them, is an important one that you should include in your answer. (It won't apply to those engineering disciplines where engineers are often doing something new that doesn't have a code.) Feb 9 '15 at 14:45

Depending on how you look at it, none and all of it.

The cycle of doing something the hard way, learning a short cut and then moving on to advanced material repeats all the way through college.

For example once I started taking Algebra, I stopped doing multiplication tables. College level math is the same way. After calculus most engineers take differential equations. At that point I really stopped doing calculus and started relying on tools to do it for me.

In controls work we use a lot of Laplace transforms to define a system. While I technically know the full theory behind the Laplace transform, I haven't done one by hand in almost a decade.

So while I haven't 'used' calculus since my 3rd-4th years of university everything I learned during them required a fundamentals of calculus.

Edit: An analogy of sorts. This is like asking someone on the 14th floor of a building how many times they use the 3rd floor. It may be never, but without the 3rd floor there wouldn't be a 14th floor either.

I agree, as discussed in a few of the other answers, that most of the time engineers do not directly use calculus (or other advanced math) very often in order to do their day to day job. And at the same time, having an understanding of it is vital for a good engineer.

I would add, though, that understanding advanced mathematics well enough to use it effectively can be extremely helpful in this present era in which advanced mathematical tools are readily available. For example, a program such as Mathcad allows the user to perform direct integration of a domain, and an engineer who understands how to use this properly can create extremely effective, accurate, and fast tools to solve routine problems.

As a geotechnical engineer, one problem I may often find myself solving in which this ability turns out to be most useful is the primary settlement $S_p$ of a soil layer. The settlement equation is simple:

$$S_p=H_{\text{layer}}\varepsilon_v=H_{\text{layer}}\frac{\Delta e}{1+e_0}$$ where $\varepsilon_v$ is vertical strain and $e$ is the void ratio of the soil.

However, it turns out that $\Delta e$ is a stress-dependent quantity, and stress varies with depth (i.e., it is a function of depth, $z$):

$$\Delta e=C_c\log{\frac{\sigma^{\prime}_0+\Delta \sigma^{\prime}}{\sigma^{\prime}_0}}$$ where $C_c$ is the compression index (constant), and $\sigma^{\prime}$ is effective stress.

(Note that in practice things are even worse since $e_0$ also varies with depth as well, but we often assume it to be constant when performing calculations in order to make things easier.)

Since $\sigma^{\prime}$ changes continuously with depth, the the usual way to do this problem is to just split the soil profile into 1 foot layers, and to use the effective stress at the center of each layer to find $S_p$ for that layer. Then you just add them up.

However, a much better, and easier, way to do this is to simply directly integrate using a tool like Mathcad! Instead of dividing a 15 foot soil column into 1 foot increments, and performing the same set of calculations at each of the 15 layers, all I have to do (a single time) is this:

1. Define pore water pressure as a function of depth, $z$ (simplest case): $$u(z)=0$$
2. Define total stress as a function of depth, $z$: $$\sigma_0(z)=\gamma_{\text{soil}}z$$
3. Define effective stress as a function of depth, $z$: $$\sigma^{\prime}_0(z)=\sigma_0(z)-u(z)$$
4. Define effective stress increase as a function of depth, $z$ (simplest case is a constant increase): $$\Delta\sigma^{\prime}(z)=1000\text{ psf}$$
5. Define change in void ratio as a function of depth, $z$: $$\Delta e(z)=C_c\log{\frac{\sigma^{\prime}_0(z)+\Delta \sigma^{\prime}(z)}{\sigma^{\prime}_0(z)}}$$

And finally, find the total consolidation of the layer down to any depth $z=H_{\text{layer}}$ by directly integrating the primary settlement equation:

$$S_p=\int_{0}^{H_{\text{layer}}}{\frac{\Delta e(z)}{1+e_0}\text{d}z}$$

This approach is quicker, more accurate, and easier than the method taught in your soil mechanics or foundations textbook. However, it requires an ability to understand and apply basic calculus in order to implement it properly.

There are loads of other examples (e.g., structural analysis of a beam in bending, groundwater flow, volumetric flow analysis of a watershed hydrograph, etc etc) in which direct integration would be a superior approach to that commonly used if the right tool is available.

An electronics engineer here, who found the maths the most difficult part of his degree.

I quite routinely have to use and manipulate complex numbers when doing RF engineering, circuit modelling and design. They have also been useful when modelling ultrasonic propagation. I have often wished that Excel handled complex numbers as a built-in type.

An understanding of ODEs is vital when designing control and feedback systems.

Understanding the concepts of Fourier series, Laplace and Z-transforms and convolution has been necessary.

The important thing for me has been in knowing what maths is out there and being able to ask a mathematician for help when needed. The mathematicians that I have consulted have invariably been delighted to help out with practical problems.

• But do you actually use Fourier series and Laplace transforms with convolution? Maybe they help you understand, but at the end of the day do you use the math? You said that you have to calculate by complex numbers, do you do this with calculus as well? Feb 10 '15 at 17:27
• @Nicholas: I have needed to know the Fourier series of a theoretical signal. I have used FFTs in signal processing. I have used Laplace less often, but textbooks on control theory are full of them. When building matching circuits I have taken S-parameters (complex reflection and transmission coefficients) off instruments, into MATLAB or a circuit simulator and done arithmetic on them. I have needed to understand the relationship between convolution and Fourier products when designing digital filters. Feb 11 '15 at 16:35

As a computational scientist, i work closely with engineers developing the software tools they use to solve different kinds of engineering problems. My work relies heavily on partial differential equations and numerical analysis, for which integrals, derivatives, taylor series, limits, green's theorem, optimization, rates of change, etc... are all the basic tools i use every day of my life.

In my opinion, professional engineers are the tool users, while i see myself as a toolmaker. An engineer can certainly use a tool without knowing much about the intricacies of how it was made... But to pick the right tool for the job at hand, you have to understand the wider variety of tools to choose from and their advantages / disadvantages. The only way to understand the advantages of one numerical tool over another, you have to understand the building blocks of that tool. For this, calculus is absolutely necessary.

I'll give an example of calculus that I used today as a Software Engineer.

We were estimating the computational time of performing an operation on each of many groups of elements. The time taken for an individual group is proportional to the size of the group squared.

We're not sure of the distribution of the sizes of the groups, but depending on different algorithms we might use, we might be able to model them as normally distributed, power-law distributed, exponentially-distributed, etc., as well as influence the parameters of those respective distributions.

Calculating the expected value of $X^2$ where $X$ is sampled from some distribution requires basic calculus knowledge :)

In general, things like this pop up from time to time. I don't know that I've ever used it explicitly in terms of writing software that carries out calculations related to calculus, nor have I used it as an authoritative decision-making tool. Usually that's left up to "try a few things and see what works best" but it's definitely useful for basic whiteboard brainstorming or estimation. In this case, it let us theorize about what kind of distribution we hope will work best, and focus our efforts on trying that path out. I can certainly say that very basic fundamentals of calculus are useful to understand the dynamics of some software systems. Four semesters is probably overkill.

• Although not strictly calculus, (and I've never used it since my 2nd year unit on algorithms), it can be useful to use proof by induction for calculating the upper and lower bounds of algorithmic complexity for a given algorithm. But if someone asked me to do it today, I'd have to Google the method to do it. Feb 10 '15 at 2:26

I have a bachelor's in computer engineering. I'm still early in my career (currently mostly software, but I'm trying to get more involved in the hardware aspect of things), but here's my experience:

I was wondering what kind of calculus do you real engineers use?

The single most-used topic for me both in school and elsewhere was the Fourier transform. It came up time and again in my electrical engineering classes, and I now work in telecommunications, where it's come up in various forms relatively often.

That said, it's the concepts and background, and understanding the physical reality through the equations that has helped me most rather than the actual numbers and calculations (which I've seen very rarely outside school). Knowing how to blindly follow rules and do calculations can help to do well in school (depending on the professor), but in my experience it's more important to have a conceptual understanding and general idea of the behavior of circuits than to be able to calculate an exact numerical answer. At work we'd get the answer the quick way -- plug the numbers into a simulator. But if you have a conceptual understanding, you'll know what to expect, and notice when something's wrong.

From my experience I'd say that the most important thing is to understand well how the equations describe the physical system and be able to translate back and forth. That is, let the equations enhance your understanding of the physical system.

Maybe you do not use anything from it, but it does enhance your mathematical reasoning, which does have a positive externality on your engineering skills?

Yes! The ability to describe a physical system in mathematical terms, and then understand and predict its behavior is a skill I gained in school, and I believe is very important for any engineer.

This is written from the point of view of someone getting a PhD in mechanical engineering. My math background is somewhat comparable to (but definitely inferior to) that of PhD students in an applied math program.

As other have indicated, the answer to this question depends greatly on the particular engineer's work. In many cases, advanced math is genuinely useless. A civil engineer mentioned code based work as an example.

As a PhD student working in computational fluid dynamics, I need a reasonably solid understanding of everything through PDEs. Math is a tool I use to solve problems, just like an experimentalist might consider a thermometer a tool. I develop mathematical models (usually solved by computers) for use by myself and other engineers.

Topics covered in my undergraduate math education that I find useful in my work:

• integral, differential, and vector calculus (Basically all of it, though I admit I've only used Lagrange multipliers once or twice since undergrad)

• probability and statistics (the class I had was fairly dumbed down, however)

• differential equations (both ordinary and partial)

I also took an undergraduate complex analysis course which I found to be fascinating, though I must admit I've used nearly none of it since then. Some of the graduate math courses I've taken and found useful include asymptotic analysis, measure-theoretic probability (not so much for the measure theory, directly, but for thinking more carefully), and numerical PDEs.

My undergrad differential equations background was fairly deficient, however. The basic ODE class must be hard to teach, because (roughly) 75% of the students in there don't need to know much about ODEs and the other 25% needs to know the subject well. (I could write a lot more on this subject, in particular, which areas I think were deficient.)

I want to go on a bit of a tangent to address a related topic. There are a large number of engineers who believe advanced math is more useless to them than it actually is, and they often are quite vocal about it. Some engineers seem to go out of their way to avoid using any sort of math at all[1], even if it would be helpful. One company that has tried to recruit people from my research group bragged that they don't do any math, as if that would entice us. To be honest, they became an inside joke. A lot of their work is code based, and while the codes tend to be conservative, they are not always correct or helpful in every case. When someone has to make an "engineering judgment", I hope the judgment is based on an evidence-based mathematical model and not speculation. (I'm not sure why this opinion about the usefulness of advanced math exists, but I think it comes partly from the difficulty of the math and also ignorance.)

Engineers who don't use advanced math should at least be aware of the potential pitfalls of blindly using engineering software based on advanced math. Many engineers trust the software as if its result is infallible. I am funded by a government agency who produces a simulation software (and I help develop the software) and I recall one of their engineers being severely annoyed at users who claim to have discovered new physics: temperatures higher than the adiabatic flame temperature (the highest temperature possible in combustion due to the first law). What actually happened was that the simulation software did not use a "TVD" scheme, and the developers assumed (perhaps implicitly) that people using the software would recognize when things go awry and add additional resolution. My impression is that they didn't want to make the software foolproof because it would slow things down dramatically, but apparently this problem cropped up so many times they added the foolproof algorithm.

This is not to say that advanced math is always necessary. While some engineers might consider it fun to overdo something with mathematical sophistication, if it's not necessary solve a problem, it's probably a waste of time.

[1] Incidentally, the same is true for programming. For a class taught my by MS advisor, he specifically designed an assignment to be "impossible" to solve in Excel because it required solution of large linear systems of equations many times. By far the easiest way to do this would be writing a few dozen lines of code. He required people to turn in their code to receive credit. He still received spreadsheets! Apparently you can do this in Excel, but you needed to type in the matrix manually! Surely not easy or fun when you need a 500x500 matrix.

If we have to answer this question very briefly, I would say:

(1) Engineers do use codes, and the applying code do not need calculus, but only calculation and software.

(2) Most engineers use codes written by others in their lifetime career.

(3) Top ones write and modify codes and software, they use math. They make the complex problems simplified for others, put them in table, software, and arithmetic formulas.

• What percentage of engineers use codes, though? Feb 11 '15 at 16:32
• @HDE226868: Any engineer that does either Design or Modeling, uses software built off of code, not necessarily the code itself.
– Paul
Feb 11 '15 at 18:38
• By "code", I mean any legal (government), industrial, or company documents such as civil codes, nautical classification, or safety regulations. I think software is for giving data, but engineers make decision based on "code". Feb 11 '15 at 18:57
• @Paul I had meant actually writing code. PdotWang - I totally misunderstood. I don't know how well this answers the question, though. Regulations aren't that much related to math. Feb 11 '15 at 19:16
• See comments from hazzey. I should mention it earlier. Sorry for the misleading. Feb 11 '15 at 19:24

The answers all generally make valid points but I think they miss the real reason engineers take a pretty standard 2 year math curriculum: efficiency in learning the rest of their coursework. The people that devised the original curricula were not interested in creating a "liberal arts" foundation where calculus would exercise your mind etc. They wanted to train engineers, plain and simple.

But to train engineers, you need to teach them subjects like mechanics, fluids, waves, etc. To learn those different topics efficiently, you need calculus and linear algebra. Sure you can replace a calculus argument by devising some very clever, elementary argument, but it's much better to give ONE argument via calculus that encompasses a variety of cases. Same thing goes for linear algebra. For example, the concept of whether the nullspace of a linear system is trivial or not ties together quite nicely with the analogous concept in linear ODEs.

One could argue all day about whether learning this way makes a better engineer or not, but one thing is clear to anyone that's taught: this is a very efficient way of training engineers. And how well one understands the math being taught will have a direct effect on how well one understands the rest of the engineering curriculum.

When I was taking courses as a "special student" at Carnegie Mellon University in Pittsburgh (in the mid-1970s),"engineering mathematics" consisted of linear algebra, ordinary and partial differential equations, and "special topics" such as power series and fourier series solutions, as well as LaPlace transforms. This is a "heavy" engineering school, and many will have programs that are "lighter."

• This does not answer the original question, mr Tom. Are you are real engineer? If so, do you use any of this calculus you learned in your profession? Feb 10 '15 at 3:54
• @NicolasBourbaki: My bio says that I have "hung around" engineers, taken courses with them, and watched what they do. So my "experience" is second hand (as an observer), rather than first hand (as an engineer). One way of characterizing my true profession is "journalist," financial, engineering, etc. Feb 10 '15 at 15:17
• You can't compare the math foundation of an engineer in the mid 70s with today. If you look at the textbooks, you can see how things have changed. Feb 12 '15 at 11:14
• @Chan-HoSuh, this is true. Some of the textbooks my father had in his undergraduate mechanical engineering courses in the early 80s are now used for graduate courses, likely because of the math. Feb 12 '15 at 17:45