Notes for MATH1050/7050 Mathematical Foundations II
Background Knowledge
The course begins with a review of elementary concepts, including Trigonometry.
1.1 Trigonometry Review
Angle Measurements: Angles are measured primarily in radians; conversion between radians and degrees can be achieved using π radians = 180 degrees.
Unit Circle: The circumference of a circle with radius 1 is 2π. An angle of 180° corresponds to π radians.
Key Concepts in Trigonometry
Trigonometric Functions:
Sine and cosine functions defined as:
If P is a point on the unit circle, OP makes an angle θ from the positive x-axis, then coordinates of P: (cos θ, sin θ).
Graphing Functions: Graphs of y = sin θ, y = cos θ can be visualized by moving around the unit circle.
Fundamental Identities:
Basic identities: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ.
Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = cos θ/sin θ.
Graphing Trigonometric Functions
Special Angles: Important values include: 0, π/6, π/4, π/3, π/2, and their respective trigonometric ratios.
CAST Rule: Quadrant-based approach to determine the sign of trigonometric functions.
Vectors
2.1 Introduction to Vectors
Vector Definition: A vector has both magnitude and direction, unlike scalars which have only magnitude.
Geometric Representation: In R2 and R3, vectors are shown as arrows; their representation depends on both length (magnitude) and direction.
Matrix Representation
Vectors can be expressed in matrix form:
2D: ( v = \begin{pmatrix} x2 - x1 \ y2 - y1 \ \end{pmatrix} )
3D: ( v = \begin{pmatrix} xQ - xP \ yQ - yP \ zQ - zP \ \end{pmatrix} )
Conversion: From geometric to matrix form and vice versa.
2.2 Addition of Vectors
Adding Vectors
Geometric Addition: Place the tail of vector w at the head of vector v to obtain their sum v + w.
Matrix Addition: For vectors ( v = \begin{pmatrix} v1 \ v2 \ \end{pmatrix} ) and ( w = \begin{pmatrix} w1 \ w2 \ \end{pmatrix} ), ( v + w = \begin{pmatrix} v1 + w1 \ v2 + w2 \ \end{pmatrix} ).
Properties of Vector Addition
Commutative: ( v + w = w + v )
Associative: ( u + (v + w) = (u + v) + w )
Identity element: ( 0 + v = v )
2.3 Scalar Multiplication of Vectors
Scalar Multiplication
Definition: Given a vector v and scalar t, ( tv ) scales the vector by |t| and possibly reverses its direction if t < 0.
Matrix Form: If ( v = \begin{pmatrix} v1 \ v2 \ \end{pmatrix} ), then ( tv = \begin{pmatrix} t \cdot v1 \ t \cdot v2 \ \end{pmatrix} ).
Unit Vector
A vector with a norm of 1 is called a unit vector, denoted as ( \hat{v} ).
2.7 The Scalar Product
Definition and Properties
The scalar (dot) product defined for two non-zero vectors v and w as: ( v \cdot w = |v| |w| cos(\theta) ), where θ is the angle between them.
Properties:
Scalar product results in a scalar value.
Orthogonal vectors yield a dot product of 0.
For any vector u and real number t, ( (tu) \cdot v = t(u \cdot v) ).
4 Complex Numbers
Introduction to Complex Numbers
Complex numbers arise from solving quadratic equations without real solutions. Represented as ( z = a + bi ), where i = ( \sqrt{-1} ).
Properties and Operations
Addition: ( z + w = (a + c) + (b + d)i )
Multiplication: Using the distributive property leading to complex numbers.
Polar Form and De Moivre's Theorem
Polar Form: A complex number can be expressed as ( z = r \text{cis } \theta ) or ( z = re^{i\theta} ).
This representation simplifies multiplication and division of complex numbers.
Powers of Complex Numbers
De Moivre's Theorem states that ( z^n = r^n \text{cis}(n\theta) ", facilitating the computation of powers of complex numbers.
Applications in Real Life
Complex numbers apply in fields such as engineering and physics for various calculations, including oscillations and waves.
Norm of a Vector
The norm (magnitude) of a vector v = (x, y) in 2-space is given by |v| = √(x² + y²). In 3-space, for v = (x, y, z), it is |v| = √(x² + y² + z²).
Component Form of a Vector
In 2-space: A vector can be expressed as v = (x, y).
In 3-space: A vector can be expressed as v = (x, y, z).
Converting Component Form to Geometric Form
To convert from component form to geometric form, visualize the vector as an arrow originating from the origin (0,0) or (0,0,0) pointing to the coordinates (x,y) in 2-space or (x,y,z) in 3-space. Conversely, to convert from geometric back to component form, determine the coordinates of the tip of the vector arrow.
Properties of the Scalar Product
The scalar (dot) product is commutative: v · w = w · v.
It is distributive: v · (u + w) = v · u + v · w.
The dot product results in a scalar value.
If vectors are orthogonal, their dot product is 0.
Scaling a vector scales the product: (tu) · v = t(u · v).
Properties of the Vector Product
The vector (cross) product is anti-commutative: v × w = - (w × v).
It is distributive: v × (u + w) = v × u + v × w.
The magnitude of the cross product is |v × w| = |v| |w| sin(θ), where θ is the angle between them.
The result of the cross product is a vector orthogonal to both v and w.
Area of a Triangle
The area can be calculated using the formula: Area = 1/2 × base × height. Alternatively, using vectors, Area = 1/2 |v × w|, where v and w are line segments of the triangle.
Torque
Torque (τ) is defined as τ = r × F, where r is the position vector and F is the force vector. Its magnitude is given by |τ| = |r| |F| sin(θ), where θ is the angle between r and F.
Vectors in Geometry
Vectors are used to represent points, directions, and geometrical shapes in space, allowing for easier computations of distances, angles, and transformations.
Forces, Displacement, Velocity, and Momentum
Force (F) is a vector that represents an interaction that changes the motion of an object. Displacement (d) is a vector that represents a change in position, Velocity (v) is displacement over time, and Momentum (p) is given by p = mv, where m is mass.
Properties of the Modulus and Complex Conjugate
The modulus of a complex number z = a + bi is given by |z| = √(a² + b²).
The complex conjugate of z, denoted as (\overline{z}), is a - bi. The product of a complex number and its conjugate yields a real number: z \overline{z} = a² + b².
Geometric Representation of Complex Numbers
Complex numbers can be represented on a plane (Argand plane) with the x-axis representing the real part and the y-axis representing the imaginary part, visualizing the number as a point or vector from the origin.
Complex Numbers in Polar Form
A complex number can be expressed as z = r \text{cis} θ or z = re^{iθ}, where r is the modulus and θ is the argument (angle).
Powers of Complex Numbers
Using De Moivre's Theorem, the nth power of a complex number in polar form is given by z^n = r^n \text{cis}(nθ), simplifying the calculation of complex exponentiation.
Adding Notes on Graphs of Trigonometric Functions
The graphs of trigonometric functions such as sine, cosine, and tangent demonstrate periodic behavior. The period of sine and cosine is 2π, while tangent has a period of π.
The amplitude of the sine and cosine functions is 1, while the tangent function approaches infinity at certain points within its period.
Secant, Cosecant, and Cotangent Functions
Secant Function: Defined as ( sec(θ) = \frac{1}{cos(θ)} ), it is the reciprocal of the cosine function. Important points include where the cosine function is zero; here, the secant function is undefined.
Cosecant Function: Defined as ( csc(θ) = \frac{1}{sin(θ)} ), it is the reciprocal of the sine function. It becomes undefined where the sine is zero.
Cotangent Function: Defined as ( cot(θ) = \frac{cos(θ)}{sin(θ)} ), it is defined where the sine function is not zero.
Geometric Representation of Vectors
Vectors are represented as arrows originating from an initial point (the tail) and terminating at an end point (the head). In a 2D space, vectors can indicate direction and magnitude from the origin to a point in the plane.
Matrix Representation of Vectors
2D Vector: A vector can be represented in matrix form as ( v = \begin{pmatrix} x \ y \end{pmatrix} ).
3D Vector: Similarly, in 3D, vectors are expressed as ( v = \begin{pmatrix} x \ y \ z \end{pmatrix} ).
Converting Between Matrix and Geometric Forms
To convert a vector from geometric form to matrix form, extract the coordinates of the vector's terminal point. Conversely, to convert from matrix form to geometric form, visualize the vector originating from the origin to the given coordinates.
Geometric Addition of Vectors
Vectors can be added geometrically by placing the tail of one vector at the head of another. This "tip-to-tail" method results in a new vector that represents their sum.
Properties of Vector Addition
Commutative: ( v + w = w + v )
Associative: ( u + (v + w) = (u + v) + w )
Identity element: ( 0 + v = v )
Geometric Scalar Multiplication
When a vector is multiplied by a scalar, its direction remains unchanged but its magnitude is scaled by the absolute value of the scalar. If the scalar is negative, the vector’s direction reverses.
Position Vectors
A position vector defines the location of a point in space relative to the origin, represented as ( r = \begin{pmatrix} x \ y \ z \end{pmatrix} ).
Norm of a Vector
The norm (or magnitude) of a vector ( v = \begin{pmatrix} x \ y \end{pmatrix} ) in 2D is given by |v| = √(x² + y²), while in 3D, ( v = \begin{pmatrix} x \ y \ z \end{pmatrix} ), it's |v| = √(x² + y² + z²).
Component Form of a Vector in 2-space and 3-space
In 2-space: A vector can be expressed as ( v = (x, y) ).
In 3-space: A vector can be expressed as ( v = (x, y, z) ).
Converting to and from Geometric and Component Form
To convert from component form to geometric form, visualize the vector as an arrow from the origin to the point defined by the components. In reverse, determine the coordinates based on the vector’s direction and length from the origin.
Properties of the Scalar Product
The scalar product is commutative: ( v \cdot w = w \cdot v ).
It is distributive: ( v \cdot (u + w) = v \cdot u + v \cdot w ).
The dot product results in a scalar.
If vectors are orthogonal, their dot product is zero.
Scaling a vector scales the product: ( (tu) \cdot v = t(u \cdot v) ).
Properties of the Vector Product
The vector product is anti-commutative: ( v \times w = - (w \times v) ).
It is distributive: ( v \times (u + w) = v \times u + v \times w ).
The magnitude of the cross product is |v \times w| = |v| |w| sin(θ).
The result of the cross product is a vector orthogonal to both ( v ) and ( w ).
Area of a Triangle
The area of a triangle can be calculated using the formula: Area = 1/2 × base × height. Alternatively, using vectors, Area = 1/2 |v \times w|, where v and w are the sides of the triangle.
Torque
Torque (τ) is defined as ( τ = r \times F ) where ( r ) is the position vector and ( F ) is the force vector. Its magnitude is given by ( |τ| = |r| |F| sin(θ) ).
Vectors in Geometry
Vectors represent points, directions, and geometrical shapes in space, making calculations of distances, angles, and transformations more manageable.
Forces, Displacement, Velocity, and Momentum
Force (F) is a vector representing an interaction that changes the motion of an object. Displacement (d) is a vector that indicates positional change, Velocity (v) is defined as displacement over time, and Momentum (p) is given by ( p = mv ), where m is mass.
Properties of the Modulus and Complex Conjugate
The modulus of a complex number ( z = a + bi ) is given by |z| = √(a² + b²).
The complex conjugate of z, denoted as ( \overline{z} ), is given by ( a - bi ). The product of a complex number and its conjugate yields a real number: ( z \overline{z} = a² + b².\n
Geometric Representation of Complex Numbers
Complex numbers are represented on a plane, called the Argand plane, where the x-axis represents the real part and the y-axis represents the imaginary part, visualizing the number as a point or vector from the origin.
Complex Numbers in Polar Form
A complex number can be expressed in polar form as ( z = r \text{cis} θ ) or ( z = re^{iθ} ), where r is the modulus and θ is the argument (angle).
Powers of Complex Numbers
Using De Moivre's Theorem, the nth power of a complex number in polar form is represented as ( z^n = r^n \text{cis}(nθ) ), which simplifies the computation of complex exponentiation.