Algebraic and Functional Foundations for Engineers
(Level 1)
You can't solve a dynamics problem until you can rearrange F = ma into a = F/m without thinking. You can't optimize a beam design until you recognize which terms in your cost function actually matter. Engineering math isn't about proving theorems. It's about manipulating symbols fast enough that they don't slow down your thinking. This level builds that fluency.
What You'll Learn
Algebraic Manipulation for Engineering Problems
In engineering, one often spends roughly half of their time in classes simply rearranging equations. At first, this may seem unnecessarily time consuming for professors to explain, but equations are rarely ever set up in a convenient format to begin with, and need to be manipulated using various techniques to extract meaningful information from them. For example, a force equation might be set up initially as √(F^2), which results in the somewhat complicated rational expression (F√(F))/√(F). In order to better understand this expression, one would need to factor it to get (F√(F))/√(F) → F, showing that, after all that work, the expression simplifies to the original variable. More complex scenarios, such as comparing the relative effectiveness of two different designs, would require one to first expand the terms of both equations in order to see which coefficients are most important.
These skills of factoring, expanding and simplifying are NOT separate topics. They are a series of moves that you can make in the game of clearing up an expression to see its structure. Each move allows you to isolate the features that you need to see in an expression. Algebraic manipulation at its best is fluently playing this series of moves.
Difference of squares: (a + b)(a - b) = a² - b²
This uses the difference of cubes formula.
Why This Matters Later
In calculus, you'll factor expressions to find limits, simplify derivatives, and integrate rational functions. In differential equations, algebraic manipulation helps isolate variables and solve characteristic equations.
Engineering Functions and Their Behavior
Linear functions model quantities with constant rates, such as velocity, force of a spring, voltage divided by resistance. As things accelerate or compound, the functions become quadratics (including projectiles and quadratic approximations for beam deflections) or exponentials (including cooling, capacitor discharge, and exponential growth or decay of a population).
Choosing the correct function family for a problem is extremely time saving. Recognising a parabola is crucial for problems involving energy, ballistics, optimisation. Similarly recognizing exponential growth is important for problems involving feedback, resonance, instability. The maths becomes just a notational vehicle and it is far more important to be able to recognised the underlying structure than to be able to work out the associated algebra.
Vertex at (-3, -4)
Why This Matters Later
Linear functions model constant-velocity motion. Quadratics appear in projectile motion and beam deflection. Exponentials describe cooling, charging circuits, and damped vibrations. Polynomials approximate complex functions via Taylor series.
Trigonometry in Engineering Analysis
Almost everything that moves, sways, or points uses trigonometry. Robot arms, vibrations, directions, AC circuits, wave interference, gear ratios and satellite orbits are all trig topics. Even the humble unit circle is no abstract geometric construct but a recipe for breaking down any amount of rotation or periodicity into forms amenable to calculation.
Identities (Pythagorean, angle sums, double angles) - these are not things you need to memorise. Once you learn what an identity is, you realise that these 4 identities are just shortcuts to simplify lengthy expressions that might otherwise take you three pages of working to simplify. You rarely ever need to explicitly state the identity for sin²θ + cos²θ in a dynamics problem - it is just faster and more efficient to write it as 1.
Four solutions: two where sin = +√2/2, two where sin = -√2/2
Why This Matters Later
Trig functions are solutions to differential equations in vibrations and waves. Fourier series decompose complex signals into sums of sines and cosines. Rotation matrices in robotics rely on trigonometric identities.
Coordinate Geometry for Engineering Applications
CAD systems, robot path planning, stress analysis. Everything in engineering happens in coordinate space. You're always asking: What's the distance between these two points? Where do two lines intersect? What's the direction perpendicular to a surface?
The distance formula, midpoint formula, and line equations may seem like "geometry review", but these formulas are used to determine clearance, centroids, and interference. Circles and parabolas occur frequently in lens design and antenna reflectors where shapes or paths need to be optimised. Having the equation of a shape helps you to calculate or adjust parameters rather than just drawing the shape.
Why This Matters Later
Coordinate geometry is essential for CAD, path planning, and kinematics. Conic sections (circles, ellipses, hyperbolas) describe orbits, reflectors, and stress distributions.
Vectors for Forces, Motion, and Work
Direction matters. Forces have direction, and so do velocities. You can't just say 10 Newtons or 5 m/s, you have to specify the direction as well. This is what vectors do, and that's not just some convenient notation - it's the only way to represent physical quantities correctly.
Vector addition (like combining forces or velocities) is a purely geometric operation, but we do it algebraically by adding the components. The dot product of two vectors gives their degree of alignment, and is useful in physical problems for calculating work (force · displacement) and for making projections of one vector on another. Physicists generally cannot do enough vector manipulation.
Why This Matters Later
Vectors are the language of forces, moments, and motion. Dot products calculate work and projections. Cross products (Level 2+) find torques and normals to surfaces.
Systems of Linear Equations in Engineering
What ever real problem you are studying will probably require you to solve more than one equation at a time. Usually the internal forces in a structure like a truss are acted on by many elements that are in balance, each joint in a truss typically giving two equations to solve. Similarly a circuit can be analyzed using Kirchhoff's laws. Every loop and node of a circuit typically provides one equation. The result is that you end up with three, five, or ten equations to be solved simultaneously.
We learn to solve systems of linear equations by either substituting and eliminating variables or in matrix form. Although these techniques are sufficient for small systems of equations (2-3 equations), as the system of equations gets larger, these methods become impractical to solve. However, the methods to solve systems of linear equations in matrix form are the very techniques that finite element analysis uses to solve a system of thousands of linear equations. The same techniques, just on a much larger scale with bigger matrices.
Why This Matters Later
Systems of equations model circuits, trusses, and fluid networks. Matrix methods scale to thousands of variables in finite element analysis. Gaussian elimination is the basis for numerical solvers.
Why Engineers Use Algebra and Functions
Engineering cannot exist without the critical foundation of algebra. To solve for the acceleration of an object (F = ma) you first need to solve for it by rearranging the equation. To optimize a design you need to determine how parameters will scale. Functions model relationships such as stress vs. strain, voltage vs. current, and flow rate vs. pressure drop. Exponential and logarithmic functions describe growth and decay as well as scaling laws that appear in everything from RC circuits to population dynamics. Algebra is not simply a group of magical formulas that get plugged into a calculator to produce an answer. Algebra is a language that we, as engineers, use to describe how the world around us behaves. If you can understand and utilize algebra you will find the underlying physics of any scenario to be much clearer.