NST Mathematics

Methods I (Michaelmas Term)

Vector algebra, coordinate systems, complex numbers, hyperbolic functions, differentiation, integration, and basic probability

• Vectors: Fundamentals

  Vector Notation

  A vector is a quantity with both magnitude and direction. Common notations:

  • Bold Roman: $\mathbf{u}$, $\mathbf{q}$ (typeset)
  • Underline/tilde: $\underline{u}$ (handwritten)
  • Arrow: $\vec{u}$ (North American convention)

  A position vector from point $A$ to point $B$ is written $\vec{AB}$. Position vectors from the origin $O$ to point $A$ are written $\vec{OA}$ or simply $\mathbf{a}$.

  ---

  Vector Addition

  Vectors add geometrically using the parallelogram rule or triangle rule:

  

  $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \quad \text{(commutative)}$

  $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \quad \text{(associative)}$

  For three non-coplanar vectors, their sum is the diagonal of a parallelepiped formed by the vectors as edges.

  

  ---

  Scalar Multiplication

  Multiplying a vector $\mathbf{a}$ by a scalar $\lambda \in \mathbb{R}$:

  $\lambda(\mathbf{a} + \mathbf{b}) = \lambda\mathbf{a} + \lambda\mathbf{b} \quad \text{(distributive)}$

  • $\lambda > 0$: stretches/compresses without changing direction
  • $\lambda < 0$: reverses direction and scales magnitude
  • $|\lambda\mathbf{a}| = |\lambda||\mathbf{a}|$

  ---

  Cartesian Components

  Right-Handed Coordinate System

  By convention, we use a right-handed system where the $x$, $y$, $z$ axes satisfy:

  • Thumb points along $x$
  • Index finger along $y$
  • Middle finger along $z$

  

  Unit Vectors

  The standard basis (orthonormal set):

  $\hat{\mathbf{i}}, \quad \hat{\mathbf{j}}, \quad \hat{\mathbf{k}} \quad \text{or} \quad \mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z$

  Properties:

  • Orthogonal: $\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{i}} = 0$
  • Unit length: $|\hat{\mathbf{i}}| = |\hat{\mathbf{j}}| = |\hat{\mathbf{k}}| = 1$

  Position Vector

  $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}} = (x, y, z)$

  Magnitude (Length)

  $|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$

  Distance Between Points

  $|\mathbf{r}_2 - \mathbf{r}_1| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

  ---

  Unit Vectors and Direction

  A unit vector has magnitude 1. To normalise any non-zero vector $\mathbf{a}$:

  $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$

  This extracts the direction while discarding magnitude information.

  ---

  Why Vectors Matter

  Vectors unify many physical quantities:

  • Displacement: position change
  • Velocity: rate of position change
  • Acceleration: rate of velocity change
  • Force: causes acceleration
  • Electric/magnetic fields: act on charges

  The operations (addition, scalar multiplication) match physical intuition: displacements add, forces combine, velocities scale.

• Scalar Product and Planes

  Definition of the Scalar Product

  The scalar product (or dot product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:

  $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$

  where $\theta \in [0, \pi]$ is the angle between the vectors.

  

  The result is a scalar (hence the name).

  ---

  Properties

  Algebraic properties:

  • Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
  • Distributive: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$
  • Self-product: $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$

  Basis vector products: $\hat{\mathbf{i}} \cdot \hat{\mathbf{i}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{k}} = 1$ $\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{i}} = 0$

  ---

  Component Form

  For $\mathbf{a} = (a_x, a_y, a_z)$ and $\mathbf{b} = (b_x, b_y, b_z)$:

  $\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z$

  This provides a computationally convenient formula derived from the basis vector properties.

  ---

  Geometric Applications

  Angle Between Vectors

  $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

  

  Perpendicular Vectors

  Two non-zero vectors are perpendicular (orthogonal) if and only if:

  $\mathbf{a} \cdot \mathbf{b} = 0$

  Component Along a Direction

  The component of $\mathbf{a}$ in the direction of a unit vector $\hat{\mathbf{n}}$:

  $a_{\parallel} = \mathbf{a} \cdot \hat{\mathbf{n}} = |\mathbf{a}|\cos\theta$

  This is the projection of $\mathbf{a}$ onto the direction $\hat{\mathbf{n}}$.

  Component Perpendicular to a Direction

  $a_{\perp} = |\mathbf{a}|\sin\theta = \sqrt{|\mathbf{a}|^2 - (\mathbf{a} \cdot \hat{\mathbf{n}})^2}$

  ---

  Equation of a Plane

  Through a Point with Given Normal

  A plane through point $\mathbf{a}$ with unit normal $\hat{\mathbf{n}}$ consists of all points $\mathbf{r}$ satisfying:

  $(\mathbf{r} - \mathbf{a}) \cdot \hat{\mathbf{n}} = 0$

  

  Why this works: the vector $\mathbf{r} - \mathbf{a}$ lies in the plane, so it must be perpendicular to the normal.

  Expanded Form

  If $\hat{\mathbf{n}} = (l, m, n)$ where $l, m, n$ are direction cosines (cosines of angles with coordinate axes):

  $lx + my + nz = p$

  where $p = \mathbf{a} \cdot \hat{\mathbf{n}}$ is the perpendicular distance from the origin to the plane.

  Through Three Points

  For three points $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ (not collinear), the plane is:

  $\mathbf{r} = \mathbf{a} + \mu(\mathbf{b} - \mathbf{a}) + \nu(\mathbf{c} - \mathbf{a})$

  The normal vector is:

  $\mathbf{n} = (\mathbf{b} - \mathbf{a}) \times (\mathbf{c} - \mathbf{a})$

  ---

  Distance from Point to Plane

  For a plane $\hat{\mathbf{n}} \cdot \mathbf{r} = p$ and point $\mathbf{q}$:

  $d = |\hat{\mathbf{n}} \cdot \mathbf{q} - p|$

  This is the signed distance (positive on one side, negative on the other).

  ---

  Why the Scalar Product Matters

  The scalar product measures alignment between vectors:

  • Maximum when vectors are parallel
  • Zero when perpendicular
  • Negative when pointing in opposite directions

  This captures the geometry of angles without explicitly computing angles, making it essential for:

  • Work done by a force: $W = \mathbf{F} \cdot \mathbf{d}$
  • Power delivered: $P = \mathbf{F} \cdot \mathbf{v}$
  • Flux through a surface: involves dot products with normals

• Vector (Cross) Product

  Definition of the Vector Product

  The vector product (or cross product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:

  $\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{\mathbf{n}}$

  where:

  • $\theta \in [0, \pi]$ is the angle from $\mathbf{a}$ to $\mathbf{b}$
  • $\hat{\mathbf{n}}$ is a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$
  • The direction of $\hat{\mathbf{n}}$ follows the right-hand rule: curl fingers from $\mathbf{a}$ to $\mathbf{b}$, thumb points along $\hat{\mathbf{n}}$

  

  The result is a vector (hence the name).

  ---

  Properties

  Algebraic properties:

  • Anti-commutative: $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$
  • Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$
  • Self-product: $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
  • Not associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$

  Geometric interpretation: $|\mathbf{a} \times \mathbf{b}| = \text{area of parallelogram formed by } \mathbf{a} \text{ and } \mathbf{b}$

  ---

  Component Form (Determinant)

  The cross product can be computed using a determinant:

  $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$

  Expanding:

  $\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y)\hat{\mathbf{i}} + (a_z b_x - a_x b_z)\hat{\mathbf{j}} + (a_x b_y - a_y b_x)\hat{\mathbf{k}}$

  ---

  Standard Basis Products

  Using the right-hand rule:

  $\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}, \quad \hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}}$

  And by anti-commutativity:

  $\hat{\mathbf{j}} \times \hat{\mathbf{i}} = -\hat{\mathbf{k}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{j}} = -\hat{\mathbf{i}}, \quad \hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}$

  ---

  Applications

  Normal Vector to a Plane

  Given two vectors in a plane, their cross product gives a normal vector:

  $\mathbf{n} = \mathbf{a} \times \mathbf{b}$

  

  Equation of a Line Through Two Points

  For points $\mathbf{a}$ and $\mathbf{b}$, the line equation is:

  $(\mathbf{r} - \mathbf{a}) \times (\mathbf{b} - \mathbf{a}) = \mathbf{0}$

  This encodes: any point $\mathbf{r}$ on the line has $\mathbf{r} - \mathbf{a}$ parallel to $\mathbf{b} - \mathbf{a}$.

  Shortest Distance: Point to Line

  Distance from point $\mathbf{q}$ to line through points $\mathbf{a}$, $\mathbf{b}$:

  $d = \frac{|(\mathbf{q} - \mathbf{a}) \times (\mathbf{b} - \mathbf{a})|}{|\mathbf{b} - \mathbf{a}|}$

  This is the magnitude of the perpendicular component.

  Rotational Motion

  The velocity of a point at position $\mathbf{r}$ rotating with angular velocity $\boldsymbol{\omega}$:

  $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$

  ---

  Triple Products

  Scalar Triple Product

  $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix} = [\mathbf{a}, \mathbf{b}, \mathbf{c}]$

  

  Properties:

  • Cyclic permutation (even): value unchanged
    • $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = [\mathbf{b}, \mathbf{c}, \mathbf{a}] = [\mathbf{c}, \mathbf{a}, \mathbf{b}]$
  • Odd permutation: sign changes
    • $[\mathbf{a}, \mathbf{c}, \mathbf{b}] = -[\mathbf{a}, \mathbf{b}, \mathbf{c}]$
  • Geometric meaning: volume of parallelepiped formed by $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$
  • Coplanar vectors: if $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = 0$, the vectors lie in the same plane

  Vector Triple Product

  $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$

  This is the BAC-CAB rule (mnemonic: “BAC minus CAB”).

  Warning: Cross product is not associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$

  ---

  Why the Cross Product Only Exists in 3D

  The cross product as defined here is specific to three dimensions. In 2D, a perpendicular “vector” would have to point out of the plane (making it essentially a scalar). In 4D and higher, the concept generalises differently (wedge product / exterior algebra).

  This is why the cross product is sometimes called the “vector product” — it produces a vector only because we happen to work in 3D.

• Coordinate Systems

  Cartesian Coordinates

  The standard Cartesian (rectangular) coordinate system uses three mutually perpendicular axes $x$, $y$, $z$ emanating from an origin $O$.

  Position vector: $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$

  Basis vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, $\hat{\mathbf{k}}$ are constant — they point in the same direction everywhere.

  ---

  Cylindrical Polar Coordinates ($r$, $\theta$, $z$)

  Useful for systems with cylindrical symmetry (pipes, wires, cylindrical objects).

  

  Coordinate Definitions

  • $r \geq 0$: perpendicular distance from $z$-axis
  • $\theta$: azimuthal angle from $x$-axis in the $xy$-plane
  • $z$: height (same as Cartesian)

  Relations to Cartesian

  $x = r\cos\theta, \quad y = r\sin\theta, \quad z = z$

  $r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\frac{y}{x}$

  Basis Vectors

  Unlike Cartesian, cylindrical basis vectors depend on position:

  $\hat{\mathbf{e}}_r = \cos\theta\,\hat{\mathbf{i}} + \sin\theta\,\hat{\mathbf{j}}$

  $\hat{\mathbf{e}}_\theta = -\sin\theta\,\hat{\mathbf{i}} + \cos\theta\,\hat{\mathbf{j}}$

  $\hat{\mathbf{e}}_z = \hat{\mathbf{k}}$

  Position vector: $\mathbf{r} = r\hat{\mathbf{e}}_r + z\hat{\mathbf{e}}_z$

  Warning: $\hat{\mathbf{e}}_r$ and $\hat{\mathbf{e}}_\theta$ change direction as $\theta$ changes.

  ---

  Spherical Polar Coordinates ($r$, $\theta$, $\phi$)

  Useful for systems with spherical symmetry (atoms, planets, central forces).

  

  Coordinate Definitions

  • $r \geq 0$: distance from origin
  • $\theta \in [0, \pi]$: polar angle from $z$-axis (inclination)
  • $\phi \in [0, 2\pi)$: azimuthal angle from $x$-axis

  Note: Physics convention differs from mathematics — physicists use $\theta$ for polar angle and $\phi$ for azimuthal, while mathematicians often swap them.

  Relations to Cartesian

  $x = r\sin\theta\cos\phi, \quad y = r\sin\theta\sin\phi, \quad z = r\cos\theta$

  $r = \sqrt{x^2 + y^2 + z^2}$

  Basis Vectors

  Again, position-dependent:

  $\hat{\mathbf{e}}_r = \sin\theta\cos\phi\,\hat{\mathbf{i}} + \sin\theta\sin\phi\,\hat{\mathbf{j}} + \cos\theta\,\hat{\mathbf{k}}$

  $\hat{\mathbf{e}}_\theta = \cos\theta\cos\phi\,\hat{\mathbf{i}} + \cos\theta\sin\phi\,\hat{\mathbf{j}} - \sin\theta\,\hat{\mathbf{k}}$

  $\hat{\mathbf{e}}_\phi = -\sin\phi\,\hat{\mathbf{i}} + \cos\phi\,\hat{\mathbf{j}}$

  Position vector: $\mathbf{r} = r\hat{\mathbf{e}}_r$

  ---

  Volume Elements

  The volume element changes with coordinate system:

  Cartesian: $dV = dx\,dy\,dz$

  Cylindrical: $dV = r\,dr\,d\theta\,dz$

  The extra factor $r$ accounts for the fact that rings at larger $r$ have larger circumference.

  Spherical: $dV = r^2\sin\theta\,dr\,d\theta\,d\phi$

  The factor $r^2\sin\theta$ comes from the Jacobian determinant. At the poles ($\theta = 0, \pi$), $\sin\theta = 0$ and the volume element vanishes — this reflects the coordinate singularity.

  ---

  Area Elements

  For surface integrals:

  Cylindrical (curved surface): $dS = r\,d\theta\,dz$

  Spherical (surface of sphere): $dS = r^2\sin\theta\,d\theta\,d\phi$

  ---

  Why Different Coordinate Systems?

  Choosing the right coordinates simplifies problems:

  Symmetry	Best Coordinates	Example
  Cylindrical	$(r, \theta, z)$	Electric field of a wire
  Spherical	$(r, \theta, \phi)$	Gravitational field of Earth
  Translational	Cartesian	Uniform field

  Key insight: The physics remains the same; only the mathematical description changes. Choose coordinates that match the symmetry to reduce complexity.

  ---

  Warning: Basis Vectors Change

  In curvilinear coordinates, derivatives of basis vectors are non-zero:

  Cylindrical: $\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta} = \hat{\mathbf{e}}_\theta, \quad \frac{\partial \hat{\mathbf{e}}_\theta}{\partial \theta} = -\hat{\mathbf{e}}_r$

  Spherical: More complex relations involving all three basis vectors.

  This means $\frac{\partial \mathbf{r}}{\partial \theta}$ is not simply $\hat{\mathbf{e}}_\theta$ — there are additional terms. Always compute vector products in Cartesian or use established formulas for operators like $\nabla$ in curvilinear coordinates.

• Complex Numbers

  Definition

  A complex number is:

  $z = x + iy$

  where:

  • $x \in \mathbb{R}$ is the real part: $\text{Re}(z) = x$
  • $y \in \mathbb{R}$ is the imaginary part: $\text{Im}(z) = y$
  • $i$ is the imaginary unit: $i^2 = -1$

  The set of all complex numbers is $\mathbb{C}$.

  ---

  Complex Conjugate

  The complex conjugate of $z = x + iy$ is:

  $\bar{z} = z^* = x - iy$

  Properties:

  • $z\bar{z} = |z|^2 = x^2 + y^2$
  • $z + \bar{z} = 2\text{Re}(z)$
  • $z - \bar{z} = 2i\text{Im}(z)$
  • $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$
  • $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$

  ---

  The Argand Diagram

  Complex numbers correspond to points in the complex plane (Argand diagram):

  • Horizontal axis: real part
  • Vertical axis: imaginary part

  

  This geometric interpretation makes complex numbers useful for 2D rotations and oscillations.

  ---

  Modulus and Argument

  Modulus (Magnitude)

  $|z| = r = \sqrt{x^2 + y^2} = \sqrt{z\bar{z}}$

  The distance from origin in the Argand diagram.

  Argument (Angle)

  $\arg(z) = \theta = \tan^{-1}\frac{y}{x}$

  Warning: Must consider the signs of $x$ and $y$ to get the correct quadrant. The principal argument satisfies $-\pi < \theta \leq \pi$.

  Use:

  • $\theta = \tan^{-1}(y/x)$ if $x > 0$
  • $\theta = \tan^{-1}(y/x) + \pi$ if $x < 0$ and $y \geq 0$
  • $\theta = \tan^{-1}(y/x) - \pi$ if $x < 0$ and $y < 0$
  • $\theta = \pi/2$ if $x = 0$ and $y > 0$
  • $\theta = -\pi/2$ if $x = 0$ and $y < 0$

  ---

  Polar Form

  Using modulus and argument:

  $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

  This is the polar form (also called modulus-argument form).

  ---

  Euler’s Formula

  $e^{i\theta} = \cos\theta + i\sin\theta$

  Derivation sketch: Expand $e^{i\theta}$, $\cos\theta$, $\sin\theta$ as power series and compare.

  Euler’s identity (when $\theta = \pi$): $e^{i\pi} + 1 = 0$

  This connects five fundamental constants: $e$, $i$, $\pi$, $1$, $0$.

  ---

  De Moivre’s Theorem

  $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

  Or equivalently: $(e^{i\theta})^n = e^{in\theta}$

  Applications

  Trigonometric identities: $\cos n\theta = \text{Re}[(\cos\theta + i\sin\theta)^n]$ $\sin n\theta = \text{Im}[(\cos\theta + i\sin\theta)^n]$

  Binomial expansion method: $\cos 3\theta = \text{Re}[(\cos\theta + i\sin\theta)^3] = \cos^3\theta - 3\cos\theta\sin^2\theta = 4\cos^3\theta - 3\cos\theta$

  Expressing trig in terms of exponentials: $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, \quad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

  ---

  Roots of Unity

  The $n$th roots of unity are solutions to $z^n = 1$:

  $z_k = e^{2\pi i k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}$

  for $k = 0, 1, 2, \ldots, n-1$.

  

  Geometric interpretation: The $n$ roots are evenly spaced around the unit circle, forming a regular $n$-gon.

  Example (cube roots): $z^3 = 1 \Rightarrow z_0 = 1, \quad z_1 = e^{2\pi i/3}, \quad z_2 = e^{4\pi i/3}$

  ---

  Roots of Complex Numbers

  To solve $z^n = w$ where $w = Re^{i\phi}$:

  $z_k = \sqrt[n]{R} \exp\left(i\frac{\phi + 2\pi k}{n}\right)$

  for $k = 0, 1, \ldots, n-1$.

  ---

  Why Complex Numbers Matter

  Complex numbers unify and simplify many areas:

  • Differential equations: exponential solutions
  • AC circuits: impedance extends resistance
  • Quantum mechanics: wave functions are complex-valued
  • Signal processing: Fourier transforms use complex exponentials
  • 2D rotations: multiplication by $e^{i\theta}$ rotates by angle $\theta$

  The algebra of complex numbers captures oscillatory and wave behaviour elegantly, replacing sines and cosines with exponentials.

• Hyperbolic Functions

  Definitions

  The hyperbolic functions are defined by analogy with trigonometric functions:

  $\cosh x = \frac{e^x + e^{-x}}{2}$

  $\sinh x = \frac{e^x - e^{-x}}{2}$

  $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

  Related functions: $\text{sech } x = \frac{1}{\cosh x}, \quad \text{cosech } x = \frac{1}{\sinh x}, \quad \coth x = \frac{1}{\tanh x}$

  ---

  Relation to Trigonometric Functions

  The connection between hyperbolic and trig functions involves the imaginary unit:

  $\cosh(ix) = \cos x$

  $\sinh(ix) = i\sin x$

  

  $\cos(ix) = \cosh x$

  $\sin(ix) = i\sinh x$

  Why the name “hyperbolic”? Like how $\cos\theta$ and $\sin\theta$ parameterise a circle $x^2 + y^2 = 1$, the functions $\cosh t$ and $\sinh t$ parameterise a hyperbola $x^2 - y^2 = 1$.

  ---

  Osborne’s Rule

  Osborne’s rule converts trigonometric identities to hyperbolic identities:

  Replace $\cos$ with $\cosh$, $\sin$ with $\sinh$. Change the sign wherever there is a product of two $\sinh$ functions.

  Examples:

  Trigonometric	Hyperbolic
  $\cos^2\theta + \sin^2\theta = 1$	$\cosh^2 x - \sinh^2 x = 1$
  $\cos(A+B) = \cos A\cos B - \sin A\sin B$	$\cosh(A+B) = \cosh A\cosh B + \sinh A\sinh B$
  $\sin(A+B) = \sin A\cos B + \cos A\sin B$	$\sinh(A+B) = \sinh A\cosh B + \cosh A\sinh B$
  $\cos 2\theta = \cos^2\theta - \sin^2\theta$	$\cosh 2x = \cosh^2 x + \sinh^2 x$

  ---

  Key Identities

  Fundamental: $\cosh^2 x - \sinh^2 x = 1$

  Addition formulae: $\cosh(A \pm B) = \cosh A\cosh B \pm \sinh A\sinh B$ $\sinh(A \pm B) = \sinh A\cosh B \pm \cosh A\sinh B$ $\tanh(A + B) = \frac{\tanh A + \tanh B}{1 + \tanh A\tanh B}$

  Double angle: $\cosh 2x = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x$ $\sinh 2x = 2\sinh x\cosh x$

  ---

  Graph Properties

  Parity:

  • $\cosh x$ is even: $\cosh(-x) = \cosh x$
  • $\sinh x$ is odd: $\sinh(-x) = -\sinh x$
  • $\tanh x$ is odd: $\tanh(-x) = -\tanh x$

  

  Key values:

  • $\cosh 0 = 1$, $\sinh 0 = 0$, $\tanh 0 = 0$

  Asymptotic behaviour:

  • $\cosh x \geq 1$ for all real $x$ (minimum at $x = 0$)
  • $\tanh x \to \pm 1$ as $x \to \pm\infty$
  • As $x \to +\infty$: $\cosh x \approx \sinh x \approx \frac{1}{2}e^x$

  ---

  Derivatives

  $\frac{d}{dx}\sinh x = \cosh x$

  $\frac{d}{dx}\cosh x = \sinh x$

  $\frac{d}{dx}\tanh x = \text{sech}^2 x$

  Compare with trig: The derivatives of $\sinh$ and $\cosh$ follow the same pattern as $\sin$ and $\cos$ but without sign changes.

  ---

  Inverse Hyperbolic Functions

  Expressed as logarithms:

  $\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1}) \quad \text{(defined for all } x\text{)}$

  $\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1}) \quad \text{(for } x \geq 1\text{)}$

  $\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \quad \text{(for } |x| < 1\text{)}$

  Derivation of $\sinh^{-1}$: If $y = \sinh^{-1} x$, then $x = \sinh y = \frac{e^y - e^{-y}}{2}$. Multiply by $e^y$, solve the quadratic in $e^y$.

  ---

  Why Hyperbolic Functions Matter

  1. Catenary: The shape of a hanging chain is $y = a\cosh(x/a)$
  2. Special relativity: Rapidity (the hyperbolic angle) uses $\tanh$
  3. Differential equations: Solutions to $\frac{d^2y}{dx^2} = y$ involve $\sinh$ and $\cosh$
  4. Complex analysis: Bridge between trigonometric and exponential functions

  Hyperbolic functions appear wherever phenomena have exponential growth/decay combined with symmetry, such as in thermal physics, fluid dynamics, and electrical transmission lines.

• Differentiation

  Leibniz Formula for Products

  For $y = u(x)v(x)$, the $n$th derivative is:

  $\frac{d^n y}{dx^n} = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}u}{dx^{n-k}} \frac{d^k v}{dx^k}$

  This generalises the product rule to higher derivatives.

  Example: If $y = x^2 e^x$, find $\frac{d^3 y}{dx^3}$.

  Let $u = x^2$ and $v = e^x$. Then:

  • $u' = 2x$, $u'' = 2$, $u''' = 0$
  • $v' = v'' = v''' = e^x$

  $y''' = u'''v + 3u''v' + 3u'v'' + uv''' = 0 + 6e^x + 6xe^x + x^2e^x = (x^2 + 6x + 6)e^x$

  ---

  Curve Sketching

  Key techniques for sketching curves $y = f(x)$:

  1. Intercepts: Where does $f(x) = 0$? Where does $y = 0$?
  2. Asymptotes:
     • Horizontal: $\lim_{x \to \pm\infty} f(x)$
     • Vertical: Where $f(x) \to \infty$
     • Oblique: Divide polynomials for rational functions
  3. Stationary points: Where $f'(x) = 0$
     • Local maximum: $f'' < 0$
     • Local minimum: $f'' > 0$
     • Point of inflection: $f'' = 0$ but derivative doesn’t change sign
  4. Symmetry: Even ($f(-x) = f(x)$) or odd ($f(-x) = -f(x)$)?
  5. Behaviour at special points: Near asymptotes, near the origin

  ---

  Newton-Raphson Method

  To find roots of $f(x) = 0$, iterate:

  $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

  

  Geometric interpretation: Follow the tangent line to the $x$-axis.

  Convergence: Quadratic (very fast) near a root. But can fail if:

  • $f'(x_n) = 0$ (horizontal tangent)
  • Starting point is poorly chosen (cycle or divergence)

  Example: Estimate $\sqrt{2}$ by solving $f(x) = x^2 - 2 = 0$.

  $x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n} = x_n - \frac{x_n}{2} + \frac{1}{x_n} = \frac{x_n}{2} + \frac{1}{x_n}$

  Starting with $x_0 = 1$: $x_1 = 1.5$, $x_2 \approx 1.417$, $x_3 \approx 1.414$.

  ---

  Implicit Differentiation

  For functions defined implicitly by $F(x, y) = 0$:

  $\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}$

  Or differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

  Example: Find $\frac{dy}{dx}$ for $x^2 + y^2 = a^2$.

  Differentiating: $2x + 2y\frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.

  ---

  Parametric Differentiation

  If $x = x(t)$ and $y = y(t)$:

  $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

  For the second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} = \frac{d}{dt}\left(\frac{y'}{x'}\right) \cdot \frac{1}{x'}$

  Example: For $x = \cos t$, $y = \sin t$:

  $\frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t$

  ---

  Logarithmic Differentiation

  For products/quotients/powers, take logarithms first:

  $\text{If } y = \frac{u^a \cdot v^b}{w^c}, \text{ then } \ln y = a\ln u + b\ln v - c\ln w$

  Differentiate: $\frac{y'}{y} = \frac{au'}{u} + \frac{bv'}{v} - \frac{cw'}{w}$

  Example: Find $\frac{d}{dx}(x^x)$.

  $y = x^x$, so $\ln y = x\ln x$. Then: $\frac{y'}{y} = \ln x + 1 \Rightarrow y' = x^x(\ln x + 1)$

  ---

  Higher Derivatives and Patterns

  Some functions have recognisable derivative patterns:

  $f(x)$	$f'(x)$	$f''(x)$	$f^{(n)}(x)$
  $e^x$	$e^x$	$e^x$	$e^x$
  $\sin x$	$\cos x$	$-\sin x$	$\sin(x + n\pi/2)$
  $\cos x$	$-\sin x$	$-\cos x$	$\cos(x + n\pi/2)$
  $\ln x$	$1/x$	$-1/x^2$	$(-1)^{n-1}(n-1)!/x^n$

• Elementary Analysis

  The Intuitive Notion of Limit

  We write $\lim_{x \to x_0} f(x) = K$ if $f(x)$ can be made arbitrarily close to $K$ by taking $x$ sufficiently close to $x_0$.

  Key point: This is about behaviour near $x_0$, not at $x_0$. The limit can exist even if $f(x_0)$ is undefined or different from $K$.

  ---

  The (ε, δ) Definition (Weierstrass)

  $\lim_{x \to x_0} f(x) = K$ if and only if: for any $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - K| < \varepsilon$ for all $x$ satisfying $0 < |x - x_0| < \delta$.

  

  Key features:

  • The order matters: for any $\varepsilon$, we must find some $\delta$
  • The condition $0 < |x - x_0|$ excludes $x_0$ itself
  • We control $x$ (through $\delta$) to control $f(x)$ (through $\varepsilon$)

  Analogy: A game where an opponent picks $\varepsilon$, and you must respond with a valid $\delta$.

  ---

  Example: Proving $\lim_{x \to 0} \frac{\sin x}{x} = 1$

  We need: for any $\varepsilon > 0$, find $\delta$ such that $|x| < \delta \Rightarrow \left|\frac{\sin x}{x} - 1\right| < \varepsilon$.

  Using the inequality $|\sin x| < |x| < |\tan x|$ for $0 < x < \pi/2$:

  • From $\sin x < x < \tan x$, we get $\cos x < \frac{\sin x}{x} < 1$
  • As $x \to 0$, $\cos x \to 1$, so $\frac{\sin x}{x}$ is squeezed to 1

  This uses the squeeze theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim g = \lim h = L$, then $\lim f = L$.

  ---

  One-Sided Limits

  Right-hand limit: $\lim_{x \to a^+} f(x)$

  Left-hand limit: $\lim_{x \to a^-} f(x)$

  The (two-sided) limit exists if and only if both one-sided limits exist and are equal.

  Example: The Heaviside step function: $H(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$

  has $\lim_{x \to 0^+} H(x) = 1$ and $\lim_{x \to 0^-} H(x) = 0$. Since these differ, $\lim_{x \to 0} H(x)$ does not exist.

  ---

  Limits at Infinity

  $\lim_{x \to \infty} f(x) = K$ if for any $\varepsilon > 0$, there exists $X$ such that $|f(x) - K| < \varepsilon$ for all $x > X$.

  Example: $\lim_{x \to \infty} \frac{1}{x} = 0$

  Given $\varepsilon > 0$, choose $X = 1/\varepsilon$. Then for $x > X$: $\left|\frac{1}{x} - 0\right| = \frac{1}{x} < \frac{1}{X} = \varepsilon$

  ---

  Continuity

  A function $f$ is continuous at $x = a$ if:

  1. $f(a)$ exists (the function is defined there)
  2. $\lim_{x \to a} f(x) = f(a)$

  (ε, δ) form: For any $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - f(a)| < \varepsilon$ for all $|x - a| < \delta$.

  Note: unlike the limit definition, this includes $x = a$.

  Types of discontinuity:

  • Removable: limit exists but ≠ function value
  • Jump: one-sided limits exist but differ
  • Essential: limit does not exist (e.g., oscillation)

  ---

  Differentiability

  A function $f$ is differentiable at $x = a$ if the limit:

  $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

  exists.

  Key relationship:

  • Differentiable $\Rightarrow$ continuous
  • Continuous $\not\Rightarrow$ differentiable

  Example where continuous but not differentiable: $f(x) = |x|$ at $x = 0$. The left and right derivatives differ.

  ---

  Big-O Notation

  We write $f(x) = O(g(x))$ as $x \to a$ if there exist constants $K > 0$ and $\delta > 0$ such that:

  $|f(x)| \leq K|g(x)| \quad \text{for all } 0 < |x - a| < \delta$

  Intuition: $f$ is bounded by a multiple of $g$ near $a$.

  Examples:

  • $x^2 + x = O(x^2)$ as $x \to \infty$ (the $x^2$ term dominates)
  • $\sin x = O(x)$ as $x \to 0$ (since $\sin x / x \to 1$)

  Notation convention: We usually state the tightest bound.

  ---

  Why Rigorous Analysis Matters

  The (ε, δ) framework:

  • Makes precise what “close” means
  • Avoids circular reasoning in limits
  • Handles pathological cases (discontinuous functions, functions that oscillate)
  • Provides foundations for calculus proofs

  Historically, calculus was developed before rigorous foundations. The (ε, δ) definition (Weierstrass, 19th century) resolved centuries of confusion about infinitesimals, convergence, and continuity.

• Taylor Series

  Taylor’s Theorem

  If $f$ has derivatives up to order $(n+1)$ near $a$, then:

  $f(a+h) = f(a) + hf'(a) + \frac{h^2}{2!}f''(a) + \cdots + \frac{h^n}{n!}f^{(n)}(a) + R_{n+1}$

  

  Lagrange remainder: $R_{n+1} = \frac{h^{n+1}}{(n+1)!}f^{(n+1)}(\xi)$

  where $\xi$ lies between $a$ and $a + h$.

  Interpretation: We can approximate smooth functions by polynomials, with the remainder giving the error.

  ---

  Maclaurin Series

  The special case of Taylor series with $a = 0$:

  $f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \cdots$

  ---

  Standard Power Series

  These series should be memorised:

  Function	Power Series	Convergence
  $e^x$	$\sum_{k=0}^{\infty} \frac{x^k}{k!}$	All $x$
  $\sin x$	$\sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}$	All $x$
  $\cos x$	$\sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$	All $x$
  $\sinh x$	$\sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!}$	All $x$
  $\cosh x$	$\sum_{k=0}^{\infty} \frac{x^{2k}}{(2k)!}$	All $x$
  $\ln(1+x)$	$\sum_{k=1}^{\infty} \frac{(-1)^{k+1} x^k}{k}$	$-1 < x \leq 1$
  $\ln(1-x)$	$-\sum_{k=1}^{\infty} \frac{x^k}{k}$	$-1 \leq x < 1$

  ---

  Binomial Expansion

  For $|x| < 1$ and any $\alpha \in \mathbb{R}$:

  $(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots$

  Special cases:

  • $\alpha = n$ (positive integer): terminates (binomial theorem)
  • $\alpha = -1$: $\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots$
  • $\alpha = \frac{1}{2}$: $\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \cdots$

  ---

  Manipulating Series

  Addition: $\sum a_n x^n + \sum b_n x^n = \sum (a_n + b_n)x^n$

  Multiplication: $(\sum a_n x^n)(\sum b_n x^n) = \sum c_n x^n$ where $c_n = \sum_{k=0}^{n} a_k b_{n-k}$

  Composition: Substitute one series into another

  Differentiation: $\frac{d}{dx}\sum a_n x^n = \sum n a_n x^{n-1}$ (within radius of convergence)

  Integration: $\int \sum a_n x^n \,dx = \sum \frac{a_n x^{n+1}}{n+1}$ (within radius of convergence)

  ---

  Radius of Convergence

  A power series $\sum a_n x^n$ converges absolutely for $|x| < R$ and diverges for $|x| > R$, where:

  $R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right| \quad \text{(ratio test)}$

  At the endpoints $x = \pm R$, convergence must be checked separately.

  ---

  Example Expansions

  Expand $\tan x$ about $x = 0$:

  Since $\tan x$ is odd, write $y = \tan x$. Taking derivatives:

  • $y(0) = 0$
  • $y' = \sec^2 x$, so $y'(0) = 1$
  • $y'' = 2\sec^2 x \tan x$, so $y''(0) = 0$
  • $y''' = 2\sec^4 x + 4\sec^2 x\tan^2 x$, so $y'''(0) = 2$

  Thus: $\tan x = x + \frac{x^3}{3} + O(x^5)$

  Expand $(1+x)\ln(1+x)$:

  Using $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$:

  $(1+x)\ln(1+x) = x + \left(1 - \frac{1}{2}\right)x^2 + \left(\frac{1}{3} - \frac{1}{2}\right)x^3 + \cdots = x + \frac{x^2}{2} - \frac{x^3}{6} + \cdots$

  ---

  Why Taylor Series Matter

  • Approximation: Truncating gives polynomial approximations
  • Limits: Evaluate indeterminate forms (L’Hôpital’s rule basis)
  • Numerical methods: Derive finite difference schemes
  • Physics: Linearisation of nonlinear equations
  • ODEs: Power series solutions

  The Taylor series is the bridge between “nice” functions (exponential, trig) and polynomials, which we can compute with.

• Series Convergence

  Definitions

  A series is $\sum_{k=0}^{\infty} u_k$. The partial sums are $S_n = \sum_{k=0}^{n} u_k$.

  Convergence: The series converges if $\lim_{n \to \infty} S_n = S$ exists (finite). We write $\sum u_k = S$.

  Formal definition: For any $\varepsilon > 0$, there exists $N$ such that $|S - S_n| < \varepsilon$ for all $n > N$.

  ---

  Necessary Condition for Convergence

  If $\sum u_k$ converges, then $\lim_{k \to \infty} u_k = 0$.

  Warning: The converse is false. Example: harmonic series $\sum \frac{1}{k}$ diverges even though $\frac{1}{k} \to 0$.

  ---

  Absolute vs Conditional Convergence

  Absolute convergence: $\sum |u_k|$ converges.

  Conditional convergence: $\sum u_k$ converges but $\sum |u_k|$ diverges.

  Key theorem: If $\sum |u_k|$ converges, then $\sum u_k$ converges.

  Riemann rearrangement theorem: Conditionally convergent series can be rearranged to converge to any value (or diverge). Absolutely convergent series are immune to rearrangement.

  ---

  Comparison Test

  For series with $0 \leq u_k \leq v_k$:

  1. If $\sum v_k$ converges, then $\sum u_k$ converges
  2. If $\sum u_k$ diverges, then $\sum v_k$ diverges

  Limit comparison: If $\lim_{k \to \infty} \frac{u_k}{v_k} = L$ with $0 < L < \infty$, then $\sum u_k$ and $\sum v_k$ behave the same.

  Example: $\sum \frac{1}{k(k+1)}$ converges since $\frac{1}{k(k+1)} < \frac{1}{k^2}$ and $\sum \frac{1}{k^2}$ converges.

  ---

  Ratio Test

  Let $L = \lim_{k \to \infty} \left|\frac{u_{k+1}}{u_k}\right|$:

  • $L < 1$: series converges absolutely
  • $L > 1$: series diverges
  • $L = 1$: test is inconclusive

  Example: $\sum \frac{k^2}{2^k}$. Ratio: $\frac{(k+1)^2/2^{k+1}}{k^2/2^k} = \frac{1}{2}\left(1 + \frac{1}{k}\right)^2 \to \frac{1}{2} < 1$. Converges.

  ---

  Harmonic Series Divergence

  $\sum_{k=1}^{\infty} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$

  

  Proof of divergence (grouping):

  $1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots$

  $> 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots$

  The partial sums grow without bound.

  ---

  Alternating Series (Leibniz Criterion)

  An alternating series $\sum (-1)^k a_k$ with $a_k > 0$ converges if:

  1. $a_k$ is monotonically decreasing for large $k$
  2. $\lim_{k \to \infty} a_k = 0$

  Example: The alternating harmonic series: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$

  This is conditionally convergent (not absolutely, since harmonic series diverges).

  ---

  Riemann Zeta Function

  $\zeta(p) = \sum_{k=1}^{\infty} \frac{1}{k^p}$

  • Converges for $p > 1$
  • Diverges for $p \leq 1$

  Special values:

  • $\zeta(2) = \frac{\pi^2}{6}$
  • $\zeta(4) = \frac{\pi^4}{90}$

  ---

  Radius of Convergence for Power Series

  For $\sum a_n x^n$, the radius of convergence is:

  $R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right| \quad \text{(if limit exists)}$

  • Converges absolutely for $|x| < R$
  • Diverges for $|x| > R$
  • At $x = \pm R$: must check separately

  Example: $\sum \frac{x^n}{n}$.

  $R = \lim \frac{1/n}{1/(n+1)} = 1$. Converges for $|x| < 1$.

  At $x = 1$: harmonic series diverges. At $x = -1$: alternating harmonic converges.

  ---

  Summary: Choosing a Test

  Situation	Preferred Test
  Comparison with known series	Comparison test
  Factorials, exponentials	Ratio test
  Alternating signs	Leibniz criterion
  Powers $1/k^p$	p-test (zeta)
  Power series	Radius of convergence

  Complexity: Ratio test is often easiest to apply. Comparison test requires knowing a suitable reference series.

• Integration Fundamentals

  The Integral as Limit of a Sum

  The Riemann integral is defined as:

  $\int_a^b f(x)\,dx = \lim_{N \to \infty} \sum_{i=1}^{N} f(x_i)\,\delta x$

  where $\delta x = (b-a)/N$ and $x_i$ are sample points in each subinterval.

  Interpretation: Area under the curve, with rectangles becoming infinitesimally thin.

  ---

  Fundamental Theorem of Calculus

  First form: If $F'(x) = f(x)$, then:

  $\int_a^b f(x)\,dx = F(b) - F(a)$

  Second form: Define:

  $F(x) = \int_a^x f(t)\,dt$

  Then $F'(x) = f(x)$.

  Significance: Differentiation and integration are inverse operations.

  ---

  Differentiation Under the Integral Sign

  For $I(q) = \int_a^b f(x, q)\,dx$:

  $\frac{dI}{dq} = \int_a^b \frac{\partial f}{\partial q}\,dx$

  If limits depend on $q$:

  $\frac{d}{dq}\int_{a(q)}^{b(q)} f(x, q)\,dx = f(b, q)\frac{db}{dq} - f(a, q)\frac{da}{dq} + \int_a^b \frac{\partial f}{\partial q}\,dx$

  ---

  Standard Techniques

  Substitution

  $\int f(g(x))g'(x)\,dx = \int f(u)\,du \quad \text{where } u = g(x)$

  Example: $\int 2x\cos(x^2)\,dx$. Let $u = x^2$, $du = 2x\,dx$:

  $\int \cos u\,du = \sin u = \sin(x^2)$

  Integration by Parts

  $\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$

  Example: $\int x e^x\,dx$.

  Let $u = x$, $dv = e^x\,dx$. Then $du = dx$, $v = e^x$:

  $\int x e^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C$

  Partial Fractions

  For rational functions, decompose into simpler fractions.

  $\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

  Solving: $1 = A(x+2) + B(x-1)$. At $x = 1$: $A = 1/3$. At $x = -2$: $B = -1/3$.

  ---

  Improper Integrals

  Type 1: Infinite limits

  $\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

  Converges if limit exists (finite).

  Type 2: Unbounded integrand

  $\int_a^b \frac{dx}{\sqrt{x-a}} = \lim_{\varepsilon \to 0^+} \int_{a+\varepsilon}^b \frac{dx}{\sqrt{x-a}}$

  Comparison test for improper integrals

  If $0 \leq f(x) \leq g(x)$ and $\int g$ converges, then $\int f$ converges.

  ---

  Symmetry and Integration

  Even function: $f(-x) = f(x)$

  $\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$

  Odd function: $f(-x) = -f(x)$

  $\int_{-a}^{a} f(x)\,dx = 0$

  

  This simplifies many integrals, especially for trigonometric and polynomial terms.

  ---

  Mean Value Theorem for Integrals

  For continuous $f$ on $[a, b]$, there exists $c \in [a, b]$ such that:

  $\int_a^b f(x)\,dx = f(c)(b - a)$

  The integral equals the function value at some point times the interval width.

  ---

  Integration by Inspection

  Recognise standard forms and their derivatives:

  Integral	Result
  $\int \frac{f'}{f}\,dx$	$\ln\|f\|$
  $\int f'(x)[f(x)]^n\,dx$	$\frac{[f(x)]^{n+1}}{n+1}$
  $\int \frac{1}{a^2 + x^2}\,dx$	$\frac{1}{a}\tan^{-1}(x/a)$
  $\int \frac{1}{\sqrt{a^2 - x^2}}\,dx$	$\sin^{-1}(x/a)$

  The key is identifying patterns that match derivative rules in reverse.

• Multiple Integrals

  Double Integrals

  The double integral over a region $R$ in the $xy$-plane:

  $\iint_R f(x, y)\,dx\,dy$

  Iterated integrals: For rectangular region $a \leq x \leq b$, $c \leq y \leq d$:

  $\iint_R f\,dx\,dy = \int_a^b \left[\int_c^d f(x, y)\,dy\right]dx$

  Order can often be swapped (Fubini’s theorem).

  ---

  Triple Integrals

  $\iiint_V f(x, y, z)\,dx\,dy\,dz$

  The triple integral gives a volume when $f = 1$.

  ---

  Change of Variables

  Under transformation $(x, y) \to (u, v)$:

  $\iint_R f(x, y)\,dx\,dy = \iint_{R'} f(x(u,v), y(u,v))\,|J|\,du\,dv$

  where the Jacobian is:

  

  $J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$

  The absolute value $|J|$ accounts for the change in area element.

  ---

  Polar Coordinates

  Transformation: $x = r\cos\theta$, $y = r\sin\theta$

  Jacobian:

  $J = \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} = r\cos^2\theta + r\sin^2\theta = r$

  Area element: $dA = r\,dr\,d\theta$

  Double integral:

  $\iint_R f\,dA = \int_{\theta_1}^{\theta_2} \int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta$

  ---

  Cylindrical Coordinates

  $(x, y, z) \to (r, \theta, z)$ with $x = r\cos\theta$, $y = r\sin\theta$, $z = z$.

  Volume element: $dV = r\,dr\,d\theta\,dz$

  Useful for: cylinders, circular symmetry about $z$-axis.

  ---

  Spherical Coordinates

  $(x, y, z) \to (r, \theta, \phi)$ with:

  • $x = r\sin\theta\cos\phi$
  • $y = r\sin\theta\sin\phi$
  • $z = r\cos\theta$

  where $r \geq 0$, $\theta \in [0, \pi]$, $\phi \in [0, 2\pi)$.

  Jacobian: $J = r^2\sin\theta$

  Volume element: $dV = r^2\sin\theta\,dr\,d\theta\,d\phi$

  Derivation:

  $J = \begin{vmatrix} \sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi \\ \cos\theta & -r\sin\theta & 0 \end{vmatrix} = r^2\sin\theta$

  ---

  Example: Volume of a Sphere

  Using spherical coordinates:

  $V = \iiint dV = \int_0^{2\pi} \int_0^\pi \int_0^R r^2\sin\theta\,dr\,d\theta\,d\phi$

  $= \left[\frac{r^3}{3}\right]_0^R \cdot \left[-\cos\theta\right]_0^\pi \cdot \left[\phi\right]_0^{2\pi}$

  $= \frac{R^3}{3} \cdot 2 \cdot 2\pi = \frac{4\pi R^3}{3}$

  ---

  Example: Gaussian Integral

  Evaluate $I = \int_{-\infty}^{\infty} e^{-x^2}\,dx$.

  Method: Square and convert to polar.

  $I^2 = \int_{-\infty}^{\infty} e^{-x^2}\,dx \int_{-\infty}^{\infty} e^{-y^2}\,dy = \iint_{-\infty}^{\infty} e^{-(x^2+y^2)}\,dx\,dy$

  Convert to polar: $x^2 + y^2 = r^2$, $dx\,dy = r\,dr\,d\theta$

  $I^2 = \int_0^{2\pi} \int_0^{\infty} e^{-r^2} r\,dr\,d\theta = 2\pi \left[-\frac{1}{2}e^{-r^2}\right]_0^\infty = \pi$

  Result:

  $\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$

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  Choosing Coordinates

  Geometry	Best Coordinates
  Rectangular box	Cartesian
  Cylinder, circular base	Cylindrical
  Sphere, central force	Spherical
  Ellipse	Modified polar or elliptical

  Match the coordinate system to the domain boundaries for simplest limits.

• Probability Theory

  Basic Definitions

  Sample space $\Omega$: set of all possible outcomes.

  Event: subset of $\Omega$.

  Probability: $P(A) \in [0, 1]$ for event $A$. Properties:

  • $P(\Omega) = 1$
  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ for disjoint events: $P(A \cup B) = P(A) + P(B)$

  ---

  Conditional Probability

  $P(A|B) = \frac{P(A \cap B)}{P(B)}$

  “Probability of $A$ given that $B$ has occurred.”

  Bayes’ theorem:

  $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

  Independence: $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$.

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  Random Variables

  A random variable $X$ is a function from $\Omega$ to $\mathbb{R}$.

  Discrete: Takes countably many values.

  Continuous: Takes values in an interval or $\mathbb{R}$.

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  Probability Distributions

  Discrete: Probability Mass Function (PMF)

  $p(x_i) = P(X = x_i)$

  Requirements: $p(x_i) \geq 0$ and $\sum_i p(x_i) = 1$.

  Continuous: Probability Density Function (PDF)

  $f(x)$ such that:

  $P(a \leq X \leq b) = \int_a^b f(x)\,dx$

  Requirements: $f(x) \geq 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$.

  ---

  Expectation Value (Mean)

  Discrete: $E[X] = \mu = \sum_i x_i p(x_i)$

  Continuous: $E[X] = \mu = \int_{-\infty}^{\infty} x f(x)\,dx$

  Linearity: $E[aX + bY] = aE[X] + bE[Y]$

  ---

  Variance and Standard Deviation

  Variance: $\text{Var}(X) = \sigma^2 = E[(X - \mu)^2] = E[X^2] - (E[X])^2$

  Standard deviation: $\sigma = \sqrt{\text{Var}(X)}$

  Properties:

  • $\text{Var}(aX) = a^2\text{Var}(X)$
  • $\text{Var}(X + c) = \text{Var}(X)$

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  Moments

  The $n$th moment is $E[X^n]$.

  The $n$th central moment is $E[(X - \mu)^n]$.

  • 1st moment = mean
  • 2nd central moment = variance
  • 3rd central moment (scaled) = skewness
  • 4th central moment (scaled) = kurtosis

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  Binomial Distribution

  $n$ independent trials, each with success probability $p$.

  $P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

  where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

  Mean: $E[X] = np$

  Variance: $\text{Var}(X) = np(1-p)$

  ---

  Poisson Distribution

  For rare events with average rate $\lambda$:

  $P(X = r) = \frac{\lambda^r e^{-\lambda}}{r!}$

  Mean: $E[X] = \lambda$

  Variance: $\text{Var}(X) = \lambda$

  Limit: Poisson is the limit of Binomial as $n \to \infty$, $p \to 0$ with $np = \lambda$ fixed.

  ---

  Normal (Gaussian) Distribution

  $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

  

  Notation: $X \sim N(\mu, \sigma^2)$.

  Mean: $\mu$

  Variance: $\sigma^2$

  Standard normal ($\mu = 0$, $\sigma = 1$):

  $f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$

  Cumulative distribution function: $\Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2}\,dt$

  ---

  Central Limit Theorem

  If $X_1, X_2, \ldots, X_n$ are independent, identically distributed random variables with mean $\mu$ and variance $\sigma^2$, then:

  $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \to N(0, 1) \quad \text{as } n \to \infty$

  where $\bar{X} = \frac{1}{n}\sum X_i$.

  Significance: Sample means tend toward normal distribution, regardless of the underlying distribution (provided it has finite variance).

  ---

  Stirling’s Approximation

  $\ln(n!) \approx n\ln n - n$

  More precisely: $n! \approx n^n e^{-n} \sqrt{2\pi n}$

  Derivation sketch: $\ln(n!) = \sum_{k=1}^{n} \ln k \approx \int_1^n \ln x\,dx = n\ln n - n + 1$

  Use in probability: Essential for approximating binomial coefficients in limiting cases.

• Methods I Tripos Questions

  This section contains worked examples from past Tripos papers relevant to Methods I topics.

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  Limits and Continuity

  Tripos 2016, Paper 2, Question 19Y

  Calculate for real $x$ and constant $a$:

  (i) $\displaystyle\lim_{x \to 0} \frac{\tan x \sin x}{\sin^3 x}$

  (ii) $\displaystyle\lim_{x \to a} \frac{\sin x - \sin a}{x - a}$

  (iii) $\displaystyle\lim_{x \to \infty} \left(\frac{x + a}{x - a}\right)^{x+a}$

  Solutions:

  (i) Using $\tan x = \frac{\sin x}{\cos x}$:

  $\lim_{x \to 0} \frac{\sin^2 x}{\cos x \sin^3 x} = \lim_{x \to 0} \frac{1}{\cos x \sin x}$

  This doesn’t exist (goes to $\infty$). Actually, let’s reconsider:

  $\frac{\tan x \sin x}{\sin^3 x} = \frac{1}{\cos x \sin^2 x}$

  Using $\sin x \sim x$ near 0: this behaves like $\frac{1}{x^2}$, so the limit is $\infty$.

  (ii) This is the derivative of $\sin x$ at $x = a$:

  $\lim_{x \to a} \frac{\sin x - \sin a}{x - a} = \cos a$

  Alternatively, use the identity $\sin x - \sin a = 2\cos\frac{x+a}{2}\sin\frac{x-a}{2}$.

  (iii) Let $y = \left(\frac{x+a}{x-a}\right)^{x+a}$. Then:

  $\ln y = (x+a)\ln\left(1 + \frac{2a}{x-a}\right)$

  For large $x$: $\ln\left(1 + \frac{2a}{x-a}\right) \sim \frac{2a}{x-a}$

  So $\ln y \sim (x+a) \cdot \frac{2a}{x-a} \to 2a$ as $x \to \infty$.

  Therefore: $y \to e^{2a}$

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  Tripos 2008, Paper 1, Question 19X

  A function is defined for integer $n \in \{0, 1, 2, 3, 4\}$:

  $\phi(x, n) = \begin{cases} x^n \sin(1/x) & x \neq 0 \\ 0 & x = 0, n \neq 4 \\ 1 & x = 0, n = 4 \end{cases}$

  (i) Determine values of $n$ for which $\phi(x,n)$ tends to a limit as $x \to 0$.

  (ii) Determine values of $n$ for which $\phi(x,n)$ is continuous at $x = 0$.

  (iii) Determine values of $n$ for which $\phi(x,n)$ is differentiable at $x = 0$.

  Solutions:

  (i) Limit exists if $n > 0$. For $n = 0$: $\sin(1/x)$ oscillates. For $n \geq 1$: $|x^n \sin(1/x)| \leq |x|^n \to 0$.

  (ii) For continuity, need $\lim_{x \to 0} \phi = \phi(0)$. The limit is 0 for $n \geq 1$, so we need $\phi(0) = 0$. This holds for $n \in \{1, 2, 3\}$.

  (iii) For differentiability:

  $\phi'(0) = \lim_{h \to 0} \frac{\phi(h) - \phi(0)}{h} = \lim_{h \to 0} \frac{h^n \sin(1/h)}{h} = \lim_{h \to 0} h^{n-1}\sin(1/h)$

  This exists (equals 0) if $n - 1 > 0$, i.e., $n \in \{2, 3\}$.

  For $n = 1$: oscillates. For $n = 0$: unbounded.

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  Vectors

  Tripos 2005, Paper 1, Question 5C

  Consider tetrahedron with vertices at $O(0,0,0)$, $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$.

  (a) Find the cosines of angles between faces.

  (b) Find the areas of the four faces.

  (c) Find the distance from $O$ to face $ABC$.

  Solution outline:

  (a) Normal to face $OAB$ is in $z$-direction. Normal to $OBC$ is in $x$-direction. These are perpendicular. Angle between faces $OAB$ and $OAC$: normals are $(0,0,1)$ and $(0,1,0)$, angle is $90°$.

  Face $ABC$ has normal $(1,1,1)$ (from cross product). Angle with other faces uses dot product.

  (b) Face $OAB$ is right triangle with area $1/2$. Similarly for $OBC$, $OAC$. Face $ABC$ has area $\frac{\sqrt{3}}{2}$.

  (c) Equation of face $ABC$: $\mathbf{r} \cdot (1,1,1) = 1$, or $x + y + z = 1$.

  Distance from $O$: $d = \frac{1}{\sqrt{3}}$.

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  Tripos 2004, Paper 1, Question 1A

  Given vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ with $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \neq 0$, define reciprocal basis $(\mathbf{a}^*, \mathbf{b}^*, \mathbf{c}^*)$ by:

  $\mathbf{a}^* = \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}$

  with analogous definitions for $\mathbf{b}^*$, $\mathbf{c}^*$.

  Show that $\mathbf{a}^* \cdot \mathbf{a} = 1$ and $\mathbf{a}^* \cdot \mathbf{b} = 0$.

  Solution:

  $\mathbf{a}^* \cdot \mathbf{a} = \frac{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})} = 1$

  $\mathbf{a}^* \cdot \mathbf{b} = \frac{(\mathbf{b} \times \mathbf{c}) \cdot \mathbf{b}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}$

  But $(\mathbf{b} \times \mathbf{c}) \cdot \mathbf{b} = [\mathbf{b}, \mathbf{c}, \mathbf{b}] = 0$ (repeated vector in scalar triple product).

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  Series

  Tripos 2014, Paper 1

  Find the expansion of $\ln(1 + x + x^2)$ for:

  (i) small $x$

  (ii) large $x$

  Solution:

  (i) $1 + x + x^2 = 1 + x + x^2$. Let $u = x + x^2$, then for small $u$:

  $\ln(1+u) = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots$

  Substitute back: $\ln(1+x+x^2) = x + x^2 - \frac{(x+x^2)^2}{2} + \cdots$

  (ii) For large $x$: $\ln(1+x+x^2) = \ln(x^2) + \ln\left(\frac{1}{x^2} + \frac{1}{x} + 1\right)$

  Let $w = \frac{1}{x} + \frac{1}{x^2}$. Then:

  $= 2\ln x + w - \frac{w^2}{2} + \cdots$