Engineering Mathematics Essentials
Algebra, trigonometry, vectors, matrices, calculus and series with worked numbers.
Algebra and equations
a x² + b x + c = 0 x = (−b ± √(b² − 4 a c)) ÷ 2a log: log(ab) = log a + log b, ln ex = x
Worked example: a quadratic
2x² − 12x + 10 = 0: discriminant = (−12)² − 4 × 2 × 10 = 64, so x = (12 ± 8) ÷ 4 giving x = 5 or x = 1. Always check by substituting back: 2 × 25 − 60 + 10 = 0.
Trigonometry
sin θ = opp ÷ hyp cos θ = adj ÷ hyp tan θ = opp ÷ adj sine rule: a ÷ sin A = b ÷ sin B cosine rule: c² = a² + b² − 2ab cos C
Worked example: a triangle with two sides and the included angle
- Sides 7 and 9 with 40° between them. Cosine rule: c = √(49 + 81 − 126 cos 40°) = 5.786.
- Sine rule: A = arcsin(7 sin 40° ÷ c) = 51.0°, so B = 180° − 40° − 51.0° = 89.0°. Area = ½ a b sin C = 20.25.
Vectors
A · B = AxBx + AyBy + AzBz = |A||B| cos θ (A × B)z = AxBy − AyBx
Worked example: force and position
F = (3, 4, 0) N acts at r = (2, 1, 0) m. |F| = 5.0 N. Moment about the origin: Mz = rxFy − ryFx = 2 × 4 − 1 × 3 = 5 N·m (counter-clockwise). The angle between them: cos θ = F · r ÷ (|F||r|) = 10 ÷ (5.0 × 2.236), θ = 26.6°.
Matrices and linear systems
A set of linear equations A x = b is solved by Gaussian elimination (or Cramer’s rule for small systems); the matrix is singular (no unique solution) when its determinant is zero. For a 2 × 2 system, x = (b₁a₂₂ − a₁₂b₂) ÷ det and y = (a₁₁b₂ − b₁a₂₁) ÷ det with det = a₁₁a₂₂ − a₁₂a₂₁. Large systems (trusses, circuits, finite elements) are solved by computer; the idea is identical.
Worked example: two equations
3x + 2y = 12 and x − y = 1: det = 3(−1) − 2(1) = -5; x = (12(−1) − 2(1)) ÷ (−5) = 3, y = (3(1) − 12(1)) ÷ (−5) = 1.8. Check: 3 × 3 + 2 × 1.8 = 12.0, and 3 − 1.8 = 1.0.
Calculus
| Function | Derivative | Integral |
|---|---|---|
| xⁿ | n xⁿ⁻¹ | xⁿ⁺¹ ÷ (n + 1) |
| eᵏˣ | k eᵏˣ | eᵏˣ ÷ k |
| sin x | cos x | −cos x |
| cos x | −sin x | sin x |
| ln x | 1 ÷ x | x ln x − x |
The derivative is the slope (rate of change); the definite integral is the area under a curve. Maxima and minima occur where the derivative is zero. The product rule (uv)′ = u′v + uv′, the chain rule and integration by parts cover most cases; numerical methods handle the rest (see Numerical Methods where available).
Worked example: a maximum and an integral
- f(x) = 12x − x³ has f′(x) = 12 − 3x² = 0 at x = 2, where f = 16; f″ = −6x < 0 there, so it is a maximum.
- ∫₀³ x² dx = 3³ ÷ 3 = 9 exactly. The trapezoid rule with three strips gives 9.5 (error 5.6 percent); Simpson’s rule with six strips gives 9.0000.
Series and approximations
sin x ≈ x − x³ ÷ 6 cos x ≈ 1 − x² ÷ 2 ex ≈ 1 + x + x² ÷ 2 (1 + x)n ≈ 1 + n x
Complex numbers
A complex number z = a + j b has magnitude √(a² + b²) and angle arctan(b ÷ a); in polar form z = |z| ∠φ. Multiplication multiplies magnitudes and adds angles. Engineers use them for AC circuits and vibration. An impedance Z = 30 + j40 Ω has magnitude 50 Ω at 53.1°; a 230 V supply then drives 4.6 A lagging by 53.1° (see Electrical Power).
Related
- Previous: Dynamics: Force, Energy and Momentum. Next: Differential Equations for Engineers.
- Programming Languages for numerical methods.
- Engineering Fundamentals: the whole guide.
FAQ
Radians or degrees?
Calculus formulas (derivatives of sin and cos, series) need radians. Set your calculator and your code accordingly.
How do I check an integral or derivative?
Differentiate your integral, or evaluate the function numerically at two close points and compare with the slope.
When do I need matrices?
Whenever you have several equations in several unknowns: trusses, circuit loops, control systems, fits and finite elements.
References
- Kreyszig, E., Advanced Engineering Mathematics, Wiley.
- Stroud, K. A., Booth, D. J., Engineering Mathematics, Red Globe Press.
- Thomas, G. B., Thomas’ Calculus, Pearson.
Last reviewed: October 2026. Worked values are computed from the stated numbers.