기록

50. Vector Integration $ \vec{A}=\vec{A}(x_{i}) $ 1) Volume Integral $ \int_{V}^{}\vec{A}dv=(\int_{V}^{}A_{1}dv, \int_{V}^{}A_{2}dv, \int_{V}^{}A_{3}dv) $ 2) Surface Integral $ \int_{S}^{}\vec{A}\cdot d\vec{a}=\int_{S}^{}\vec{A}\cdot \hat{n}da=\int_{S}^{}\sum_{i}^{}A_{i}da_{i} $ (Outward Normal is positive.) 3) Line Integral $ \int_{BC}^{}\vec{A}\cdot d\vec{s}=\int_{BC}^{}\sum_{i}^{}A_{i}dx_{i} ..

35. Gradient Operator (Vector Differential Operator) Scalar Function $ \phi (x_{1},x_{2},x_{3}) $ $ \phi' (x_{1}',x_{2}',x_{3}')=\phi (x_{1},x_{2},x_{3}) $: Coordinate Transformation $ \frac{\partial \phi '}{\partial x_{i}'}=\sum_{j}^{}\frac{\partial \phi }{\partial x_{j}}\frac{\partial x_{j}}{\partial x_{i}'} $: Chain Rule $ x_{j}=\sum_{k}^{}\lambda _{kj}x_{k}' $: Inverse Coordinate Transformat..

33. Angular Velocity Instantaneous Axis of Rotation Angular Velocity: $ \omega =\frac{d\Theta }{dt}=\dot{\Theta } $ Velocity: $ v=R\frac{d\Theta }{dt}=R\omega $ $ \dot{\vec{r}}=\vec{v} $ $ R=r\sin \alpha $ $ v=R\omega =r\omega \sin \alpha $ $ \Rightarrow \vec{v}=\vec{\omega }\times \vec{r} $ 34. Infinitesimal Rotation Infinitesimal Rotation can be represented by vector. Finite Rotation cannot be..

29. Rectangular Coordinate Position: $ \vec{r}=\vec{r}(t) $ Velocity: $ \vec{v}\equiv \frac{d\vec{r}}{dt}=\dot{\vec{r}} $ Acceleration: $ \vec{a}\equiv \frac{d\vec{v}}{dt}=\frac{d^{2}\vec{r}}{dt^{2}}=\ddot{\vec{r}} $ Rectangular Coordinates (Unit Vector가 시간에 대한 상수) Position: $ \vec{r}=x_{1}\hat{e_{1}}+x_{2}\hat{e_{2}}+x_{3}\hat{e_{3}}=\sum_{i}^{}x_{i}\hat{e_{i}} $ Velocity: $ \vec{v}=\dot{\vec{r..

27. Differentiation of Scalar Function Scalar Function $ \phi =\phi (s) $ in the $ x_{i},x_{i}' $ coordinates, $ \phi =\phi ', s=s' \Rightarrow d\phi =d\phi ', ds=ds' $ $ \frac{d\phi }{ds}=\frac{d\phi '}{ds'}=(\frac{d\phi }{ds})' $ 28. Differentiationn of Vector Function Vector Function $ \vec{A} $ w.r.t. a scalar S $ A_{i}'=\sum_{j}^{}\lambda _{ij}A_{j} $ $ \frac{dA_{i}'}{ds'}=\frac{d}{ds'}\sum..

23. Vector Product(Cross Product) $ \vec{C}=\vec{A}\times \vec{B} $ $ C_{i}=\sum_{j,k}^{}\epsilon_{ijk}A_{j}B_{k} $ Levi-Civita Density(Permutation Symbol): $ \epsilon_{ijk}=\left\{\begin{matrix} 0 \\ +1 \\ -1 \end{matrix}\right. $ 0: two or three indexes are same ($ \epsilon_{112},\epsilon_{113},\epsilon_{121},\epsilon_{131},\epsilon_{211},\epsilon_{311} $) ($ \epsilon_{221},\epsilon_{223},\eps..

21. Position Vector $ \vec{A}=(x_{1},x_{2},x_{3}) $ $ \vec{B}=(\bar{x_{1}},\bar{x_{2}},\bar{x_{3}}) $ $ |\vec{A}|=\sqrt{\vec{A}\cdot \vec{A}}=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=\sqrt{\sum_{i}^{}x_{i}^{2}} $: Distance from the Origin to the Point $ (x_{1},x_{2},x_{3}) $ $ \sqrt{\sum_{i}^{}(x_{i}-\bar{x_{i}})^{2}}=\sqrt{(\vec{A}-\vec{B})\cdot(\vec{A}-\vec{B})}=|\vec{A}-\vec{B}| $: Distance from ..

19. Scalar Product(Dot Product) $ \vec{A}\cdot \vec{B}=\sum_{i}^{}A_{i}B_{i}=AB\cos (\vec{A},\vec{B}) $ $ |\vec{A}|=\sqrt{A_{1}^{2}+A_{2}^{2}+A_{3}^{2}}\equiv A $: Magnitude(Length) of $ \vec{A} $ $ \frac{A_{i}}{A}=\Lambda_{i}^{A}, \frac{B_{i}}{B}=\Lambda_{i}^{B} $: Direction Cosines of $ \vec{A} $ and $ \vec{B} $ $ \cos \Theta =\cos (\vec{A},\vec{B})=\cos \alpha \cos \alpha'+\cos \beta+\cos \be..

17. Definitions of a Scalar and a Vector in Terms of Transformation Properties $ x_{i}'=\sum_{j}^{}\lambda _{ij}x_{j} $의 좌표변환($ \sum_{j}^{}\lambda _{ij}\lambda _{kj}=\delta _{ik} $) $ \phi $ is unaffected → scalar $ A_{i}'=\sum_{j}^{}\lambda _{ij}A_{j} $ is transformed from the $ x_{i} $ system to the $ x_{i}' $ system → vector 18. Properties of Scalar and Vector $ \vec{A}, \vec{B} $: Vector, $ ..

13. Rotating Coordinate Axes 1) Counterclockwise about the $ x_{3} $-axis $ x_{1}'=x_{2}, x_{2}'=-x_{1}, x_{3}'=x_{3} $ $ \lambda _{11}=\lambda _{13}=\lambda _{22}=\lambda _{23}=\lambda _{31}=\lambda _{32}=0 $ $ \lambda _{12}=\cos (x_{1}',x_{2})=1 $ $ \lambda _{21}=\cos (x_{2}',x_{1})=-1 $ $ \lambda _{33}=\cos (x_{3}',x_{3})=1 $ $ \Rightarrow \vec{\lambda_{1}} =\bigl(\begin{smallmatrix} 0 & 1 & ..