Mathematics & Surveying
Engineering mathematics fundamentals and surveying principles for civil engineers
Table of Contents
1. Algebra
Quadratic Equations
x = (-b ± √(b² - 4ac)) / 2a
For equation ax² + bx + c = 0
Discriminant (D = b² - 4ac)
- D > 0: Two real distinct roots
- D = 0: One real repeated root
- D < 0: Two complex conjugate roots
Sum & Product of Roots
- Sum: x₁ + x₂ = -b/a
- Product: x₁ · x₂ = c/a
Sequences and Series
Arithmetic Progression (AP)
- nth term: aₙ = a₁ + (n-1)d
- Sum: Sₙ = n(a₁ + aₙ)/2
- Sum: Sₙ = n[2a₁ + (n-1)d]/2
- d = common difference
Geometric Progression (GP)
- nth term: aₙ = a₁ · r^(n-1)
- Sum (r ≠ 1): Sₙ = a₁(1 - rⁿ)/(1 - r)
- Infinite (|r| < 1): S∞ = a₁/(1 - r)
- r = common ratio
Logarithms & Exponents
Logarithm Properties
- log(xy) = log x + log y
- log(x/y) = log x - log y
- log(xⁿ) = n log x
- logₐb = ln b / ln a
- logₐa = 1, logₐ1 = 0
Exponent Properties
- aᵐ · aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1, a⁻ⁿ = 1/aⁿ
- a^(1/n) = ⁿ√a
Complex Numbers
- Rectangular form: z = a + bi (where i² = -1)
- Polar form: z = r(cos θ + i sin θ) = r∠θ
- Magnitude: |z| = √(a² + b²)
- Argument: θ = tan⁻¹(b/a)
- Euler's: e^(iθ) = cos θ + i sin θ
2. Trigonometry
Fundamental Identities
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Reciprocal Identities
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ = cos θ/sin θ
Triangle Solutions
Law of Sines
a/sin A = b/sin B = c/sin C
Use when: ASA, AAS, or SSA (ambiguous)
Law of Cosines
c² = a² + b² - 2ab·cos C
Use when: SAS or SSS
Triangle Area Formulas
- Base-Height: A = ½bh
- Two sides + included angle: A = ½ab·sin C
- Heron's Formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Sum & Difference Formulas
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Double & Half Angle Formulas
Double Angle
- sin 2A = 2 sin A cos A
- cos 2A = cos²A - sin²A
- cos 2A = 2cos²A - 1 = 1 - 2sin²A
- tan 2A = 2tan A/(1 - tan²A)
Half Angle
- sin(A/2) = ±√[(1 - cos A)/2]
- cos(A/2) = ±√[(1 + cos A)/2]
- tan(A/2) = sin A/(1 + cos A)
3. Calculus
Differentiation Rules
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec²x |
Differentiation Techniques
Product Rule
(fg)' = f'g + fg'
Quotient Rule
(f/g)' = (f'g - fg')/g²
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Integration
| Function | Integral |
|---|---|
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | -cos x + C |
| cos x | sin x + C |
Applications of Calculus
- Rate of Change: dy/dx at a point gives instantaneous rate
- Optimization: Set f'(x) = 0, check f''(x) for max/min
- Area under curve: ∫[a,b] f(x)dx
- Volume of Revolution: V = π∫[a,b] [f(x)]²dx (disk method)
- Arc Length: L = ∫[a,b] √(1 + [f'(x)]²)dx
4. Differential Equations
First Order ODEs
Separable Equations
dy/dx = f(x)g(y)
Solution: ∫dy/g(y) = ∫f(x)dx
Linear First Order
dy/dx + P(x)y = Q(x)
Integrating factor: μ = e^∫P(x)dx
Second Order Linear ODEs
ay'' + by' + cy = 0 (Homogeneous)
Characteristic Equation: ar² + br + c = 0
- Distinct real roots (r₁, r₂): y = C₁e^(r₁x) + C₂e^(r₂x)
- Repeated root (r): y = (C₁ + C₂x)e^(rx)
- Complex roots (α ± βi): y = e^(αx)(C₁cos βx + C₂sin βx)
5. Leveling
Leveling is the process of determining the difference in elevation between points on the earth's surface.
Key Formulas
Height of Instrument (HI)
HI = Elevation + BS
BS = Backsight (reading on known point)
Unknown Elevation
Elev = HI - FS
FS = Foresight (reading on unknown point)
Leveling Terms
| Term | Description |
|---|---|
| Backsight (BS) or (+) | Rod reading on point of known elevation |
| Foresight (FS) or (-) | Rod reading on point of unknown elevation |
| Turning Point (TP) | Point where instrument is moved; both BS and FS taken |
| Benchmark (BM) | Permanent reference point with known elevation |
| Intermediate Foresight (IFS) | Rod reading on points between setups |
Error Checking
- Arithmetic Check: ΣBS - ΣFS = Last Elev - First Elev
- Allowable Error: E = k√D (where D = distance in km, k = constant)
- Typical k values: 8-12 mm for ordinary leveling
6. Traverse Computation
Latitude & Departure
Latitude (North-South)
L = D × cos(Bearing)
+N = North, -S = South
Departure (East-West)
D = D × sin(Bearing)
+E = East, -W = West
Closure & Precision
- Error in Latitude: ΣL ≠ 0 (should close)
- Error in Departure: ΣD ≠ 0 (should close)
- Linear Error of Closure: LEC = √(ΣL² + ΣD²)
- Relative Precision: 1/n where n = Perimeter/LEC
- Bearing of Error: tan⁻¹(ΣD/ΣL)
Traverse Adjustment Methods
Compass Rule (Bowditch)
Correction proportional to line length
CL = (L/P) × Error in L
CD = (L/P) × Error in D
Transit Rule
Correction proportional to latitude/departure
CL = (|L|/Σ|L|) × Error in L
CD = (|D|/Σ|D|) × Error in D
7. Area Calculations
DMD Method (Double Meridian Distance)
- First line: DMD = Departure of first line
- Subsequent: DMD = Previous DMD + Dep(prev) + Dep(current)
- Area: 2A = Σ(DMD × Latitude)
Coordinate Method
2A = Σ[Xᵢ(Yᵢ₊₁ - Yᵢ₋₁)]
or equivalently:
2A = Σ(XᵢYᵢ₊₁ - Xᵢ₊₁Yᵢ)
Numerical Integration
Trapezoidal Rule
A = (d/2)(y₁ + 2y₂ + 2y₃ + ... + yₙ)
d = common interval
Simpson's Rule
A = (d/3)(y₁ + 4y₂ + 2y₃ + 4y₄ + ... + yₙ)
n must be odd (even number of intervals)
8. Curve Surveying
Simple Circular Curves
Curve Elements
- R: Radius
- Δ (or I): Intersection/Central angle
- T: Tangent distance
- L: Length of curve
- E: External distance
- M: Middle ordinate
- C: Long chord
Formulas
- T = R tan(Δ/2)
- L = (πRΔ)/180 = RΔ (rad)
- C = 2R sin(Δ/2)
- E = R[sec(Δ/2) - 1]
- M = R[1 - cos(Δ/2)]
- D = 1145.916/R (arc)
Compound & Reverse Curves
- Compound Curve: Two or more simple curves in the same direction with common tangent
- Reverse Curve: Two simple curves turning in opposite directions
- Common tangent: Shared tangent at Point of Compound/Reverse Curvature (PCC/PRC)
Spiral/Transition Curves
Provides gradual transition from tangent to circular curve
- Spiral Length: Ls = V³/(RC) where C = rate of increase of centripetal acceleration
- Spiral Angle: θs = Ls/(2R) radians
- Used for: Highways and railways for smooth transition
Key Takeaways for the Board Exam
- ✓Leveling: HI = Elev + BS; Elev = HI - FS
- ✓Traverse: Latitude = D cos(bearing); Departure = D sin(bearing)
- ✓Precision: 1/n where n = Perimeter/Error of Closure
- ✓Law of Sines: ASA, AAS; Cosines: SAS, SSS
- ✓Curve tangent: T = R tan(Δ/2)
- ✓Simpson's Rule: Requires odd number of ordinates
- ✓Quadratic: x = (-b ± √(b²-4ac))/2a
- ✓Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
Mathematics Topics
AlgebraTrigonometryCalculusDifferential EquationsLevelingTraverse ComputationArea CalculationsCurve Surveying