Physics and Mathematics

1. In terms of magnitudes and angle: $\vec{a}\cdot\vec{b} = ab\cos\theta$ where a and b are magnitudes of vectors $\vec{a}$ and $\vec{b}$, and $\theta$ is the angle between them.

2. In terms of components: $\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z$ where $a_x, a_y, a_z$ and $b_x, b_y, b_z$ are the components of vectors $\vec{a}$ and $\vec{b}$ along the x, y, and z axes.

Note: The dot product equals zero for perpendicular vectors since cos(90°) = 0.

The cross product $\vec{a} \times \vec{b}$ is:

Magnitude: $|\vec{a} \times \vec{b}| = ab\sin\theta$ where $\theta$ is the smaller angle between vectors

Direction: Perpendicular to the plane containing both vectors, following the right-hand thumb rule (fingers along $\vec{a}$, curl toward $\vec{b}$, thumb gives direction)

In component form: $\vec{a} \times \vec{b} = (a_y b_z - a_z b_y)\hat{i} + (a_z b_x - a_x b_z)\hat{j} + (a_x b_y - a_y b_x)\hat{k}$

Key properties:

• $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$ (non-commutative)
• The cross product of parallel vectors is zero
• In a right-handed coordinate system: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$

The two primary methods of vector addition are:

1. Triangle Rule:

   • Place the tail of the second vector at the head of the first vector
   • The resultant vector extends from the tail of the first vector to the head of the second vector
   • Represents sequential application of displacements/forces

2. Parallelogram Rule:

   • Draw both vectors from a common origin
   • Complete the parallelogram with these vectors as adjacent sides
   • The diagonal from the common origin represents the resultant vector
   • Represents simultaneous application of displacements/forces

Both methods yield identical results mathematically: if a⃗ + b⃗ = c⃗, the resultant vector c⃗ will have the same magnitude and direction regardless of which method is used.

From rectangular components $a_x$ and $a_y$:

Magnitude calculation:

• $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$

Direction calculation:

• $\theta = \tan^{-1}\left(\frac{a_y}{a_x}\right)$ (angle with positive x-axis)

For three-dimensional vectors with components $a_x$, $a_y$, and $a_z$:

• $|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$

Example: If a vector has components $a_x = 3$ and $a_y = 4$, its magnitude is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$, and its direction is $\theta = \tan^{-1}(4/3) \approx 53.1°$ from the positive x-axis.

The right-hand thumb rule is used to determine the direction of the cross product of two vectors.

Key points: • It provides a systematic way to find the perpendicular direction of a vector resulting from a cross product • In physics, this rule helps determine directions of:

• Magnetic fields from currents
• Angular momentum
• Torque

Steps to apply:

1. Orient your right hand with fingers pointing in direction of first vector
2. Curl your fingers toward the direction of second vector
3. Your extended thumb indicates the perpendicular direction of the cross product

Example: When determining the direction of torque (τ = r × F), point fingers along position vector (r), curl toward force vector (F), and thumb shows torque direction.

The cross product is anti-commutative, meaning vector order is critical:

• Mathematically: b⃗ × a⃗ = -(a⃗ × b⃗) • The direction of b⃗ × a⃗ is exactly opposite to the direction of a⃗ × b⃗

To determine the direction of b⃗ × a⃗ when you know a⃗ × b⃗:

1. Reverse/invert the direction of a⃗ × b⃗
2. Or use the right-hand rule again but with vectors in reverse order

This anti-commutative property has important implications in physics: • Changing the order of vectors in angular momentum calculations reverses the rotation direction • In electromagnetic equations, reversing current direction reverses the magnetic field direction

Note: This contrasts with dot products, which are commutative (a⃗ · b⃗ = b⃗ · a⃗)

The derivative is defined as:

$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$

Where:

• Δy is the change in the dependent variable
• Δx is the change in the independent variable
• The limit represents the instantaneous rate of change

This represents the slope of the tangent line to a function at a specific point, rather than the average rate of change over an interval.

Example: For the function f(x) = x², the derivative is found by: $f'(x) = \lim_{\Delta x \to 0} \frac{(x+\Delta x)^2 - x^2}{\Delta x} = \lim_{\Delta x \to 0} \frac{2x\Delta x + (\Delta x)^2}{\Delta x} = 2x$

Secant lines are fundamental to understanding derivatives:

• A secant line passes through two points (x, f(x)) and (x+Δx, f(x+Δx)) on a curve
• The slope of this secant line equals Δy/Δx = [f(x+Δx) - f(x)]/Δx
• As Δx approaches zero, the secant line approaches the tangent line
• The slope of this tangent line is precisely the derivative dy/dx

This geometric interpretation shows why the derivative represents the instantaneous rate of change - it's the limit of average rates of change over increasingly smaller intervals.

Note: When visualizing this process, imagine a secant line pivoting around a fixed point until it becomes tangent to the curve as the second point gets arbitrarily close to the first.

A local maximum of a function f(x) occurs at x = x₀ when:

1. The first derivative equals zero: f'(x₀) = 0
2. The second derivative is negative: f''(x₀) < 0

This means:

• The slope of the tangent line at x₀ is zero (tangent is horizontal)
• The function is concave downward at that point
• The function value f(x₀) is greater than values at nearby points

Example: For f(x) = -x² + 4x - 3, the maximum occurs at x = 2 where f'(2) = 0 and f''(2) = -2 < 0

Key integration rules:

Trigonometric functions:

• $\int \sin x\,dx = -\cos x + C$
• $\int \cos x\,dx = \sin x + C$
• $\int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C$
• $\int \cot x\,dx = \ln|\sin x| + C$
• $\int \sec x\,dx = \ln|\sec x + \tan x| + C$
• $\int \csc x\,dx = \ln|\csc x - \cot x| + C$

Other common forms:

• $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
• $\int \frac{1}{x}\,dx = \ln|x| + C$
• $\int \frac{1}{x^2+a^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$
• $\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\frac{x}{a} + C$
• $\int \sec^2 x\,dx = \tan x + C$
• $\int \csc^2 x\,dx = -\cot x + C$