Engineer sees clay and wants to make a brick. Artist sees clay and wants to make a sculpture. Architects do the both.
Dare to think differently.
Integral equations have integration boundaries open, unspecified. That's the point your creativity, as an engineer, inventor should kick in.
You can not use amorphous blob of matter, put it through physics equations of aerodynamics and obtain an airfoil. It does not work that way.
Politicians can not solve your integral or differential equation but they will tell you what the solution should be.
You spend years in school to learn how to quantify an invention, without enough focus to inquire how invention is made at the first place.
The amorphous blob of matter is used to explain governing equations in aerodynamics. But it is not explained how the blob became an airfoil.
It is always more interesting what is said (calculated) than which underlying language grammar (mathematical) rules are used.
If you link Maxwell's equations, magnetic flux, EM induction, motor torque, speed with electric tattoo machine, physics lecture will rock!!
Surface and volume integrals should be explained using tattoos. They are a good example for arbitrary surface and ink volume calculation.
I think my goal is to make student understand 100% what math is about and only then to say "I don't like it".
Interesting career path: AirAsia Airliner Pilot (First Officer, right seat in cockpit) is model and Miss Thailand (2005), Chananporn Rosjan.
Length of a musical note as a mathematical property has way less significance than emotional perception of the sound (note) of that length.
Britney Spears way to create integers (as oppose to Peano Axioms): "Baby, One More Time".
In many fields you have to prove why certain calculations are necessary. In math you can just say "I can do that because axioms said I can".
In physics, we can quantify things without knowing what they are and where they come from. That's apparently called knowledge.
Most of the physics is developed using an equation for which we don't know where it comes from and why it is there. Schroedinger's Equation.
Once you start defining your own initial, boundary conditions for differential equations in physics you've entered world of real creativity.
Once you start defining your own path, volume, surface, time interval, of integration, in physics, you've entered world of real creativity.
Strict, precise Newton's Law of Gravitation does not prevent you to enter into it a completely random number for mass or distance.
While a proof, in math, has to be very logical and precise, the genesis of it is usually described as art or unexplainable inspiration.
Different arguments what belongs to a certain set, and why, are not part of math. They are things called disciplines (physics, gambling..).
It's puzzling how math can accept any given set cardinality, yet it takes so much argument to say what belongs to a set at the first place.
If you were to invent mathematics, possibly from scratch, what would you invent first? What would be the situation to start it? Which rules?
Some time ago I borrowed from a library a computer science book. Beside the problem about Dinning Philosophers somebody wrote "Why??".
You can't apply math to real life like you apply paint to the wall. But you can use math way of thinking for some real life situations.
You can supply to math any number that comes first to your mind. Math will count how many are there of each. And that's prob. distribution.
Math does not care where the numbers are coming from. It's like a cashier in supermarket. No matter what's in basket there always be a sum
Functions in math should be introduced to students as pairs of numbers (arbitrary, to start with), and not as algebraic formulae.
From math point of view, calculus could've come from simple "Let's assume that...". It was not necessary to analyze a physical process.
Some parts of mathematics relationships are developed by quantifying completely non mathematical relationships.
Coulomb did not have any idea where the electrical charges are coming from or what they are, yet he formulated powerful Coulomb's Law
Quantifying something does not mean you explained it. Newton did not have the slightest idea what gravity is yet he defined Law of Gravity.
Axiomatic systems, as a way of thinking, need not to be a part of mathematics only.
Knowledge of physics laws are only necessary but not sufficient condition to be a good, innovative engineer. Innovation is an art.
Inventors play with initial conditions, final solution ranges for force, movements, speed, and not with exact solutions of PDEs in between.