26 topics
Applied math help, from a word problem to a working model
Applied math hands you the messy half of the problem: a paragraph about a cooling cup of coffee or a fish population, with no equation anywhere in sight. The technique you eventually need is usually one you already own from calculus. The part nobody taught you is the translation — choosing the variables, deciding what is changing with respect to what, and working out which real detail you are allowed to throw away.
Where students get stuck
I can solve the differential equation, I cannot get it out of the paragraph
Translate one clause at a time and write nothing else until you are done. Rate of change of becomes a derivative on the left. Proportional to becomes a constant k times whatever is named next. The difference between the temperature and the room becomes that subtraction in brackets. Newton’s law of cooling is that sentence transcribed and nothing more. Do the transcription before any algebra, and label every symbol with its units in the margin, because a term whose units do not match its neighbours is a mistranslation you can catch in seconds rather than after two pages.
Which method does this differential equation want?
Sort by shape before reaching for a technique. If you can get every y onto one side and every t onto the other, separate and integrate. If it is first-order linear — y prime, plus some function of t times y, equals some function of t — use the integrating factor. If it is second order with constant coefficients, write the characteristic equation, and when there is a forcing term on the right, add a particular solution guessed from the form of that term. Checking those three in that order takes a minute and covers most of a first course.
My numerical answer drifts away from the exact one
It is supposed to, and the real question is whether it drifts at the rate it should. Euler’s method is first order: halve the step and the error roughly halves. Classical Runge–Kutta is fourth order: halve the step and the error falls by about sixteen. So never judge one run in isolation — run it twice at different step sizes and compare. If halving the step barely changes the answer, you have converged. If it changes everything, either the step is far too large or the equation is stiff, and an explicit method will keep failing however small you go.
How am I supposed to know whether my model is any good?
Three checks, before you trust a single number that comes out of it. Units: every term being added has to carry the same ones, and if they do not the model is wrong no matter how clean the arithmetic looks. Limiting cases: push a parameter to zero or to infinity and see whether you get the obvious answer, since no fishing pressure ought to return the unfished population. Order of magnitude: work out roughly what to expect on the back of an envelope first, so a result a thousand times too large announces itself. In applied math a lot of the marks live in this reasoning.
What's covered
Applied Math topics you can work through with a tutor, generate practice on, or turn into flashcards and a study plan.
Building a model
- Turning a written scenario into equations
- Choosing variables, parameters and units
- Dimensional analysis
- Stating and defending assumptions
- Validating and interpreting results
Differential equations
- Separable equations
- First-order linear equations and integrating factors
- Second-order linear equations with constant coefficients
- Systems of differential equations
- Equilibria and stability
- Slope fields and phase lines
Multivariable and vector calculus
- Partial derivatives
- Gradients and directional derivatives
- Double and triple integrals
- Line and surface integrals
- Divergence and curl
Numerical methods
- Root finding by bisection and Newton’s method
- Numerical integration by trapezoid and Simpson’s rule
- Euler and Runge–Kutta methods
- Error, convergence and stability
- Interpolation and curve fitting
Optimisation and data
- Constrained optimisation and Lagrange multipliers
- Linear programming and the simplex method
- Least squares regression
- Matrix methods for large systems
- Discrete versus continuous modelling choices
Applied Math questions
Is this the engineering mathematics course, or the applied stream in a math department?
The groups above cover both. Engineering versions weight differential equations, vector calculus and numerical methods; applied math streams add more modelling and optimisation and expect you to justify the assumptions. Name your course and it will stay inside that syllabus instead of wandering into the other one.
My assignment is a modelling project, not a problem set. Can it help without taking it over?
Yes, and that is the better use of it. Talk through the scenario out loud, argue about which effects to keep, then let it poke holes in the assumptions the way a marker will. The model has to be yours, but the sanity checking is exactly the part students skip and lose marks on.
Can it look at my code when the simulation gives nonsense?
Share your screen and it reads your code and your output and talks you through where the method is going wrong — a step size too large, an index off by one, a boundary condition never applied. It does not run the code itself, so you stay the one executing and checking it.
Can it explain a phase plane, rather than just describing one?
It sketches on the whiteboard while it talks, so nullclines, equilibria and the trajectories curving around them get drawn as they are being explained. Stability is one of those topics that stays abstract in words and becomes obvious the moment the arrows are on the page.
Stuck on applied math right now?
Talk it through out loud, share your screen, and watch it worked out step by step on a whiteboard.
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