Review

Dot Product

Physicist's Dot Product: (dot product gives the angle) for $\overrightarrow{u}$ and $\overrightarrow{v}$ with $0 \leq \theta \leq \pi$ as the angle, then $\overrightarrow{u} \cdot \overrightarrow{v} = \|\overrightarrow{u}\| \|\overrightarrow{v}\| \cos{\theta}$

• dot product: the length of projected vector times the length of base vector. (could be negative if pointing opposite dirrection)

• $|A|\cos{\theta}$ is the length of $A$ projected to $|B|$

Property of Dot Product

1. $u \cdot v = v \cdot u$
2. $u \cdot (v + w) = u \cdot v + u\cdot w$
3. $C(u \cdot v) = (Cu) \cdot v = u \cdot (Cv)$
4. $v \cdot v = \|v\|^2$

Law of Cosines: $C^2 = A^2 + B^2 - 2AB\cos{c}$ Scalar Projection: $|comp_{\overrightarrow{u}}(v)| = \|proj_{\overrightarrow{u}}(v)\| = \frac{|\overrightarrow{u}\cdot \overrightarrow{v}|}{\|\overrightarrow{u}\|}$ Projection of Vectors: projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ is $proj_{\overrightarrow{u}}(v) = (\cos{\theta} \cdot \|v\|) \cdot \frac{\overrightarrow{u}}{\|u\|} = \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\|\overrightarrow{u}\|^2}\cdot \overrightarrow{u}$

Cross Product

Cross Product ($\times$): perpendicular ($\perp$) to all input vectors

• obtained by solving system of equation of $\overrightarrow{w} \cdot x = 0 | \forall x$ input (only particular solution are cross product, because there might be potentially infinite many solutions)

• $\overrightarrow{w} = \overrightarrow{u} \times \overrightarrow{v} = \det \begin{pmatrix} \overrightarrow{i} \overrightarrow{j} \overrightarrow{k}\\ u_1 u_2 u_3\\ v_1 v_2 v_3\\ \end{pmatrix}$

• there are (at least) two vectors perpendicular to all input, let the first one be $x$, second one be $y$, by right hand rule, cross product must be in dirrection of $z$, which means cross product is not commutative

Formal Determinant: $<span class="arithmatex"><span class="MathJax_Preview">\det \begin{pmatrix} \overrightarrow{i} \overrightarrow{j} \overrightarrow{k}\\ u_1 u_2 u_3\\ v_1 v_2 v_3\\ \end{pmatrix} = \overrightarrow{i} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_3}\\ \overrightarrow{v_2} \overrightarrow{v_3}\\ \end{pmatrix} - \overrightarrow{j} \det \begin{pmatrix} \overrightarrow{u_1} \overrightarrow{u_3}\\ \overrightarrow{v_1} \overrightarrow{v_3}\\ \end{pmatrix} + \overrightarrow{k} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_2}\\ \overrightarrow{v_2} \overrightarrow{v_2}\\ \end{pmatrix}</span><script type="math/tex">\det \begin{pmatrix} \overrightarrow{i} \overrightarrow{j} \overrightarrow{k}\\ u_1 u_2 u_3\\ v_1 v_2 v_3\\ \end{pmatrix} = \overrightarrow{i} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_3}\\ \overrightarrow{v_2} \overrightarrow{v_3}\\ \end{pmatrix} - \overrightarrow{j} \det \begin{pmatrix} \overrightarrow{u_1} \overrightarrow{u_3}\\ \overrightarrow{v_1} \overrightarrow{v_3}\\ \end{pmatrix} + \overrightarrow{k} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_2}\\ \overrightarrow{v_2} \overrightarrow{v_2}\\ \end{pmatrix}$

Determinant: for 2 by 2, $\det \begin{pmatrix} a b\\ c d\\ \end{pmatrix} = ad - bc$

Properties of Cross Product:

• Anticommutative: $\overrightarrow{u} \times \overrightarrow{v} = -(\overrightarrow{v} \times \overrightarrow{u})$

• Distributive: $\overrightarrow{u} \times (\overrightarrow{v} + \overrightarrow{w}) = \overrightarrow{u} \times \overrightarrow{v} + \overrightarrow{u} \times \overrightarrow{w}$

• Multiplication by Constant: $c(\overrightarrow{u} \times \overrightarrow{v}) = (c \overrightarrow{u}) \times \overrightarrow{v} = \overrightarrow{u} \times (c \overrightarrow{v})$

• Zero Vector / Itself: $\overrightarrow{u} \times \overrightarrow{0} = \overrightarrow{0} \times \overrightarrow{u} = 0 = \overrightarrow{v} \times \overrightarrow{v}$

• Scalar Triple Product: $\overrightarrow{u} \cdot (\overrightarrow{v} \times \overrightarrow{w}) = (\overrightarrow{u} \times \overrightarrow{v}) \cdot \overrightarrow{w}$

Angle: $\sin(\theta) = \frac{\|\overrightarrow{u} \times \overrightarrow{v}\|}{\|\overrightarrow{u}\|\|\overrightarrow{v}\|}$

Area of Parallelgram: $\|\overrightarrow{u} \times \overrightarrow{v}\|$ ($\overrightarrow{u} \times \overrightarrow{v}$ is positive if $\overrightarrow{u}$ rotate in positive dirrection to get $\overrightarrow{v}$, $\overrightarrow{i} \times \overrightarrow{j}$ is positive. Since cross product is determinant is area.)

Torque: $\tau = \overrightarrow{r} \times \overrightarrow{F}$ where $\overrightarrow{r}$ is the radius and $\overrightarrow{F}$ is the force.

• $\tau = \pm\|\overrightarrow{r}\| \|\overrightarrow{F}\| \sin{\theta}$ (positive if spin in positive dirrection)

Line & Plane

Line: a point $p = (x_0, y_0, z_0)$ and a dirrection vector $\langle{a, b, c}\rangle$, then a line is $\begin{cases} x = x_0 + at\\ y = y_0 + bt\\ z = z_0 + ct\\ \end{cases}$ or $\langle{x, y, z}\rangle = \langle{x_0, y_0, z_0}\rangle + t\langle{a, b, c}\rangle$

• or you can solve for $t$, then $\begin{cases} t = \frac{x - x_0}{a}\\ t = \frac{y - y_0}{b}\\ t = \frac{z - z_0}{c}\end{cases}$

• symmetric equation: $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$

Plane: $\pm \overrightarrow{n} \cdot \overrightarrow{PQ} = 0$

Angle between Two Plane: Angle $(0 < \theta < \frac{\pi}{2})$ between $P_1$ and $P_2$ with normal vectors $n_1$ and $n_2$ is $\cos{\theta} = \frac{|\overrightarrow{n_1} \cdot \overrightarrow{n_2}|}{\|n_1\|\|n_2\|}$

Distance between a Point and Line & Plane

Distance Between Line $(Q, \overrightarrow{v})$ and point $P$ is: $\frac{\|\overrightarrow{QP} \times \overrightarrow{v}\|}{\|\overrightarrow{v}\|}$

Distance between Plane $(Q, \overrightarrow{n})$ point $P$ is: $d = \|\text{proj}_n{\overrightarrow{QP}}\| = |\text{comp}_n \overrightarrow{QP}| = \frac{|\overrightarrow{QP} \cdot n|}{\|n\|}$

Conic Section

Vector Valued Function

Limit: limit exists if both side limit exists Continuous: limit exists and equal function value

Dot Product of Vector Valued Function: $\overrightarrow{r_1} \cdot \overrightarrow{r_2} = f_1f_2 + g_1g_2 + h_1h_2$

• Dot Product Derivative $(\overrightarrow{r_1} \cdot \overrightarrow{r_2})' = \overrightarrow{r_1} \cdot \overrightarrow{r_2}' + \overrightarrow{r_1}' \cdot \overrightarrow{r_2} = f_1f_2' + f_1'f_2 + g_1g_2' + g_1'g_2 + h_1h_2' + h_1'h_2$

Cross Product Derivative: $(\overrightarrow{r_1}(t) \times \overrightarrow{r_2}(t))' = \overrightarrow{r_1}(t) \times \overrightarrow{r_2}'(t) + \overrightarrow{r_1}'(t) \times \overrightarrow{r_2}(t)$

Arc Length

In $\mathbb{R}^2$, $\text{Arc Length} = \int_{x=a}^{x=b} \sqrt{(f'(x)^2 + 1)} dx$ In $\mathbb{R}^3$, $\begin{align} \text{Arc Length} &= \int_{t=a}^{t=b} \sqrt{(f'(t))^2 + (g'(t))^2 + (h'(t))^2} dt\\ &= \int_{t=a}^{t=b} \|\overrightarrow{r}'(t)\| dt\\ \end{align}$

Arc Length Function: function computes arc length from $t=a$ to $t=t$ given $\overrightarrow{r}(t)$ is $\S(t) = \int_a^t \|\overrightarrow{r}'(u)\| du$.

$\overrightarrow{T}(t) = \frac{\overrightarrow{r}'(t)}{\|\overrightarrow{r}'(t)\|} = \frac{\overrightarrow{r}'(t)}{\|\frac{dS}{dt}\|}$

Curvature: $\kappa(S) = \|\frac{d \overrightarrow{T}}{dS}\| = \|\overrightarrow{T'}(S)\|$ for $\overrightarrow{r}(S)$ (parameterized by arc length $S$)

• $\kappa(t) = \frac{\|\overrightarrow{T'}(t)\|}{\|\overrightarrow{r'}(t)\|}$

• In $\mathbb{R}^3$: $\kappa(t) = \frac{\|\overrightarrow{r}'(t) \times \overrightarrow{r}''(t)\|}{\|\overrightarrow{r}'(t)\|^3} = \frac{\|\overrightarrow{v} \times \overrightarrow{a}\|}{\|\overrightarrow{v}\|^3}$, which means $\|\overrightarrow{T}'(t)\| = \frac{\|\overrightarrow{r}'(t) \times \overrightarrow{r}''(t)\|}{\|\overrightarrow{r}'(t)\|^2}$

• In $\mathbb{R}^2$: $\kappa(x) = \frac{|\frac{d^2y}{dx^2}|}{(1+(\frac{dy}{dx})^2)^{\frac{3}{2}}} = \frac{|y''(x)|}{\text{Arc Length}^3}$

$\overrightarrow{v} \cdot \overrightarrow{w} = 0 \implies \|\overrightarrow{v} \times \overrightarrow{w}\| = \|\overrightarrow{v}\|\|\overrightarrow{w}\|$

Principal Unite Normal Vector: $\overrightarrow{N}(t) = \frac{\overrightarrow{T}'(t)}{\|\overrightarrow{T}'(t)\|}$

• always point in the same direction as curvature

Binormal Vector: $\overrightarrow{B}(t) = \overrightarrow{T}(t) \times \overrightarrow{N}(t)$

Acceleration Components on Curve:

• $a_{\overrightarrow{T}} = \frac{\overrightarrow{T} \cdot \overrightarrow{a}}{\|\overrightarrow{T}\|} = \overrightarrow{T} \cdot \overrightarrow{a}$

• $a_{\overrightarrow{N}} = \frac{\overrightarrow{N} \cdot \overrightarrow{a}}{\|\overrightarrow{N}\|} = \overrightarrow{N} \cdot \overrightarrow{a} = \frac{\|\overrightarrow{v} \times \overrightarrow{a}\|}{\|\overrightarrow{v}\|} = \sqrt{\|a\|^2 - a_{T}^2}$

$|F_{\text{centripital}}| = \frac{mv^2}{r}$

Multivariable

Level Curve: level curve at $z=c$ is the curve in $R^2$ of $z = f(x, y)$. Contor Map: several labeled level curves on same coordinate axes (ie. multiple level curves in the same dirrection) Vertical Trace: making one independent (instead of dependent) variable constant

If $z = f(x, y)$, the limit of $f(x, y)$ as $(x, y)$ approaches $(a, b)$ is $C$, denoted $\lim_{(x, y) \rightarrow (a, b)} f(x, y) = C$ provided that for all $\epsilon$ so that when $0 < \text{dist}((x, y), (a, b)) < S$, $|f(x, y) - C| < S$

Limits and Continuity

Limit in 3D: if $0 \leq \text{distance}((x, y), (a, b)) < \delta \implies 0 \leq |f(x, y)-L| < \epsilon$ the limit of $z = f(x, y)$ is $\lim_{(x, y) \rightarrow (a, b)} f(x, y) = L$

Find the Limit when Exists

1. try to plug in
2. try to simplify, cancel denominators, plug in
3. If get $\frac{0}{0}$, show $\lim_{(x, y) \rightarrow (a, b)} f(x, y)$ does not exists
4. $\frac{\text{something}}{0}$ means derivative DNE

Prove Limit DNE:

1. Write arbitrary $x, y, z, ...$ all in terms of polynomials of one variable, say $x$.
2. Find some polynomials $X_1, X_2, Y_1, Y_2$ such that $\lim_{(X_1, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow \neq \lim_{(X_2, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow$ and none of them are $\frac{0}{0} \uparrow$