Math Review

Chantilly Physics Club 11/9/22

Vectors

Definition

A vector is a directed line segment. It is a quantity with both magnitude and direction.

We typically write vectors in two ways:

Component form: $\vec{a} = \langle x, y, z \rangle$
Cartesian basis form: $\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}$

Operations on Vectors

Basic Operations

The magnitude of a vector, denoted $\|\vec{a}\|$ , is found by the Pythagorean theorem:

\|\vec{a}\| = \sqrt{x^2 + y^2}

Vector addition and subtraction occur component-wise:

\vec{a} + \vec{b} = \langle x_1 + x_2, y_1 + y_2 \rangle

Scalar multiplication:

\lambda \vec{a} = \langle \lambda x, \lambda y \rangle

The angle of a 2D vector:

\theta = \arctan \frac{y}{x}

Component form from magnitude and angle:

\vec{A} = \| \vec{A} \| \cos \theta \, \hat{i} + \| \vec{A} \| \sin \theta \, \hat{j}

Vector Products

Dot Product

\vec{A} \cdot \vec{B} = \|\vec{A}\| \|\vec{B}\| \cos \theta

Or component-wise:

\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y

Cross Product

\|\vec{A} \times \vec{B}\| = \|\vec{A}\| \|\vec{B}\| \sin \theta

\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = \begin{vmatrix} A_y & A_z \\ B_y & B_z \end{vmatrix} \hat{i} - \begin{vmatrix} A_x & A_z \\ B_x & B_z \end{vmatrix} \hat{j} + \begin{vmatrix} A_x & A_y \\ B_x & B_y \end{vmatrix} \hat{k}

Where

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Calculus

Limits

The limit of a function $f(x)$ as $x$ approaches $c$ is $L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ (but $x \neq c$ ). Denoted:

\lim_{x \to c} f(x) = L

If no such value exists, the limit does not exist.

Differentiation

The derivative of $f$ at $x=a$ is:

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

Common derivative rules:

\frac{d}{dx} c = 0 \quad \frac{d}{dx} x^n = n x^{n-1} \quad \frac{d}{dx} e^x = e^x \quad \frac{d}{dx} \ln x = \frac{1}{x}

Sum, difference, scalar, product, and quotient rules:

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

\frac{d}{dx}[cf(x)] = c f'(x)

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Trigonometric derivatives:

\frac{d}{dx} \sin x = \cos x \quad \frac{d}{dx} \cos x = -\sin x \quad \frac{d}{dx} \tan x = \sec^2 x

Chain rule:

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Integration

Definite integral (area under the curve):

\int_a^b f(x)\,dx

Properties:

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

\int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx

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

\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx

Indefinite integral / anti-derivative:

\int f(x)\,dx = F(x) + C \quad \text{where } F'(x) = f(x)

Basic rules:

\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad \int e^x dx = e^x + C \quad \int \frac{1}{x} dx = \ln x + C