Vector Operations
\begin{align*} &\vec{u} =< u_1 ,u_2 ,u_3> \\ &\vec{v} =< v_1 ,v_2 ,\ v_3> \\ &|\vec{u} |=\sqrt{u_1^2 +u_2^2 +u_3^2}\\ &\hat{u} =\frac{\vec{u}}{|\vec{u}|}\\ &\vec{u} +\vec{v} =< u_{1} +v_{1} ,u_{2} +v_{2} ,u_{3} +v_{3} >\\ &\vec{u} -\vec{v} =\vec{u} +( -\vec{v}) =< u_{1} -v_{1} ,u_{2} -v_{2} ,u_{3} -v_{3} >\\ &\end{align*}
Dot Product (Evaluate works, Perp test)
\begin{align*} \vec{u} \cdotp \vec{v} &=|\vec{u} |\ |\vec{v} |\ cos( \theta ) \\ &=u_{1} v_{1} +u_{2} v_{2} +u_{3} v_{3} \ ( Cosine\ Rule) \end{align*}
Cross Product
Generate a vector perp to both two vectors, Parallel test, Evaluate Area
\vec{u} \times \vec{v} =\ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3} \end{vmatrix}\\
\begin{align*} |\vec{u} \times \vec{v} | &=|\vec{u} |\ |\vec{v} |\ sin( \theta )\\ &=\text{Area\ of\ Parallelogram}\\ &=\text{2 Times Area of Triangle} \end{align*}
Box Product
Projection and Scalar Projection
$$ \begin{aligned} &Proj_{\vec{u}}\vec{v}=|\vec{v}|cos(\theta)\widehat{u}=|\vec{v}|\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\frac{\vec{u}}{|\vec{u}|}=\frac{\vec{u}\cdot\vec{v}}{|\vec{u}|^2}\vec{u}\\ &Scal_{\vec{u}}\vec{v}=|\vec{v}|\cos(\theta)=|\vec{v}|\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}=\frac{\vec{u}\cdot\vec{v}}{|\vec{u}|}\\ &|Proj_{\vec{u}}\vec{v}|(maginititude)=|Scal_{\vec{u}}\vec{v}| (absvalue) \end{aligned}$$
Line, Plane, Curve and Surface Equations
Plane
\begin{align*} &\vec{n} \ =\ < a,b,c >\ ( normal\ vector) ,\ P=( x_{0} ,y_{0} ,z_{0}) ,\ Q=( x,y,z)\\ &\overrightarrow{PQ} \ =\ < x-x_{0} ,y-y_{0} ,\ z-z_{0} >\ ( a\ vector\ in\ the\ plane)\\ &\vec{n} \cdotp \overrightarrow{PQ} =0\ ( orthognal)\\ &\Longrightarrow Dot-Normal\ Form:\ a( x-x_{0}) +b( y-y_{0}) +c( z-z_{0}) =0\ \\ &\Longrightarrow General\ Form:\ ax+by+cz=d\ ( d\ is\ a\ constant) \ \\ \end{align*}
Line
Take a line as the intersection of two planes ( general form) \begin{cases} a_{1} x+b_{1} y+c_{1} z+d_{1} =0\\ a_{1} x+b_{1} y+c_{1} z+d_{1} =0 \end{cases}
Parameterization of line (dot-direction form)
\vec{v} = < a,b,c > \text{(parallel to the line)}\\ r( t) =\ < x_{0} +at,\ y_{0} +bt,\ z_{0} +ct >
Surface
Explicit Form: z=f( x,y) Implicit Form: F(x,y,z)=0 \vec{r}( t) =\, < x( u,v) ,y( u,v) ,z( u,v)>
Curve
See curve as the intersection of two surfaces \begin{align*} \begin{cases} F( x,y,z) =0\\ G( x,y,z) =0 \end{cases}\\ \vec{r}( t) =\, < x(t) ,y(t) ,z(t)> \end{align*}
Arc Length; (Unit) Tangent Vector, (Unit) Normal Vector, Binormal Vector; Curvature and Torsion
Arc Length: s= \int _{a}^{t} |\vec{r} '( t) |\ dt=\int _{a}^{b} |\vec{r} '( t) |\ dt
Unit Tangent Vector: \vec{T}( t) =\frac{\vec{r} '( t)}{|\vec{r} '( t) |}
Unit Normal Vector: \vec{N}( t) =\frac{\vec{T} '( t)}{|\vec{T} '( t) |} Unit Binormal Vector: \vec{B}( t) =\vec{T} \times \vec{N}
Curvature: \kappa =\left| \frac{d\vec{T}}{ds}\right|
Torsion: \begin{align*} &\frac{d\vec{B}}{ds} =\frac{d(\vec{T} \times \overrightarrow{N)}}{ds}\\ &=\frac{d\vec{T}}{ds} \times \vec{N} +\vec{T} \times \frac{d\vec{N}}{ds}\\ &=\ \kappa \vec{N} \times \vec{N} +\vec{T} \times \frac{d\vec{N}}{ds} \ \ by\ \vec{N} =\frac{\frac{d\vec{T}}{ds}}{|\frac{d\vec{T}}{ds} |} =\frac{1}{\kappa }\frac{d\vec{T}}{ds} \Longrightarrow \frac{d\vec{T}}{ds} \\ &=\kappa \vec{N}\\ &=\vec{T} \times \frac{d\vec{N}}{ds} \ \ by\ \kappa \vec{N} \times \vec{N} =0\ ( parallel)\\ &\Longrightarrow \frac{d\vec{B}}{ds} =\vec{T} \times \frac{d\vec{N}}{ds}\\ &\Longrightarrow \\ &( 1) \ \frac{d\vec{B}}{ds} \perp \vec{T} \ by\ cross\ product\\ &( 2) \ \frac{d\vec{B}}{ds} \perp \vec{B} \ by\ |\vec{B} |=1( constant) \Longrightarrow \vec{B} '\cdotp \vec{B} =0\ like\ tangent\ line\ of\ circle\perp radius\\ &\Longrightarrow \frac{d\vec{B}}{ds} //\ \vec{T} \times \vec{B}\\ &\Longrightarrow \frac{d\vec{B}}{ds} //\ \vec{N} \ \ {\displaystyle by\ \ \vec{T} \times \vec{B} \ //\ \vec{N}}\\ &\Longrightarrow \frac{d\vec{B}}{ds} =-\tau \vec{N} \ ( negative\ by\ convention)\\ &\Longrightarrow \ -\frac{d\vec{B}}{ds} \cdotp \vec{N} =\tau \vec{N} \cdotp \vec{N}\\ &\Longrightarrow \tau =\ -\frac{d\vec{B}}{ds} \cdotp \vec{N} \ \ by\ \vec{N} \cdotp \vec{N} \ =1\\ &\Longrightarrow |\tau |=\left| -\frac{d\vec{B}}{ds} \cdotp \vec{N}\right| =\left| \frac{d\vec{B}}{ds}\right| |\vec{N} |\ cos( \theta ) \ \xRightarrow[|\vec{N} |=1]{\theta =0}\left| \frac{d\vec{B}}{ds}\right| \\ \end{align*} We call \kappa curvature, \tau torsion.
Partial Derivative, Chain Rule, Level Curve, Gradient
Crtical Points, Local Min/Max (Point), Saddle Point, Abs Min/Max, Lagrange Multiplier Crtical Points: \ f_{x} =f_{y} =0\ or\ f_{x} \ DNE\ \ or\ f_{y} \ DNE
Vector Field, Line Integral, Surface Integral
Potential Functions and Conservative
If exist function \phi such that \vec{F} = \nabla \phi , we say function \vec{F} is conservative, and call \phi the potential function of gradient vector field \vec{F}
Conservative Discriminator \vec{F} =\, < f,\ g,\ h >, If f_{y} =g_{x} and f_{z} =h_{x} and $ g_{z} =h_{y}$, then vector field vec{F} is conservative
Fundamental Thm of Line Integral If \vec{F} = < f, g, h > is conservative, then Circulation=\oint _{a}^{b}\vec{F} \cdotp \vec{T} \ ds=\phi ( b) -\phi ( a)\\
Definition of \nabla Operator \nabla \ :=\ < \frac{\partial \ }{\partial x} ,\ \frac{\partial \ }{\partial y} ,\ \frac{\partial \ }{\partial z} >\\
Definition of Curl and Divergence Curl(\vec{F}) = \nabla \times \vec{F} Divergence(\vec{F}) =\nabla \cdotp \vec{F}
Green’s Thm ( special case of Stoke’s Thm,when the surface is flat)
Circulation= \oint _{C}\vec{F} \cdotp \vec{T} \ ds=\iint _{R} Curl\ \vec{F} \ dA\ \ ( Curl\ Form)\\
Flux= \oint _{C}\vec{F} \cdotp \vec{N} \ ds\ =\iint _{R} Divergence\ \vec{F} \ dA\ ( Divergence\ Form)\\
Stoke’s Thm
\begin{align*} &\oint _{C}\vec{F} \cdotp \vec{T} \ ds\ ( circulation\ across\ boundary)\\ &=\iint _{S} Curl(\vec{F}) \cdotp \vec{N} \ dA\ ( surface\ intergal)\\ &=\iint _{S} Curl(\vec{F}) \cdotp \left(\overrightarrow{t_{u}} \times \overrightarrow{t_{v}}\right) \ dA\\ \end{align*}
Divergence Thm
\oiint_S \vec{F} \cdotp \vec{N} \ dS=\iiint _{D} Div(\vec{F}) \ dV