UTS 33130 Mathematical Modelling 1
Summary:
The notes cover a Mathematical Modelling course with topics including 2D and 3D coordinate systems, describing lines and planes, scalars and vectors, trigonometry formulas, vector components, dot product, cross product, equations of lines and planes, differential calculus, numerical methods, complex numbers, polar coordinates, logarithmic differentiation, integrals, differential equations, matrices, determinants, Cramer’s Rule, linear system of equations, sequences, series, and power series. The notes provide a thorough review of the material covered in the course and are intended to prepare students for the final exam.
Excerpt:
Mathematical Modelling
2D coordinate system
- (x, y) ∈ ℝ2
- The distance between two points A(x1, y1 ) and B(x2 ,y2) is d = √(x2 – x1)2 + (y2 – y1)2
3D coordinate systems
- Use the righthand rule to locate the third axis
- (x, y, z) ∈ ℝ3
3D coordinate systems: distance
- Distance between two points in space; P1 (x1, y1, z1) and P2 (x2, y2, z2) be two points in space, is given by: d = √(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
2D and 3D: Describing Lines and planes
Equation | 2D | 3D |
y = 2 | A line | A plane |
𝑥 = 3, 𝑦 = 4 | A point | A line (z not defined) |
x2 + y2 = 16 | Circle | Circular cylinder |
- Equation of a sphere with center C (h, k, l) and radius r: (x – h)2 + (y -k)2 + (z – l)2 = r2
Scalars and Vectors
- Scalars: have magnitude, not the direction
- Vectors: have BOTH magnitude and direction
- Displacement: change of position of an object or particle relative to the starting position
Trigonometry Formulas
Right Triangle | General Triangle | Exact Values |
sinθ = a/c
cosθ = b/c tansθ = a/b Pythagoras’ Theorem: c2 = a2 + b2 Area of a triangle: 1/2ab |
Sine rule: a/sinθa = b/sinθb = c/sinθc
Cosine rule: a2 = b2 + c2– 2bc cosθa Area of a triangle: 1/2bc sinθa |
Vector Components
- x and y components: xy
- The length/magnitude of each component is:
- x = cosθ
- y = sinθ, where |A| is the length of
- = (Ax, Ay) for 2D
- = (Ax, Ay, Az) for 3D
Reviews