접벡터,tangent_vector (rev. 1.13)
tangent vector는:
접선,tangent_line과 유사? (curr. goto there)
접평면,tangent_plane?
see also MIT_Multivariable_Calculus#s-6
Bazett ¶
곡선 위의 한 점의 위치벡터,position_vector  "$\vec{r}(t)$")
가정.
를 어떻게 구할 것인가? 우선
-\vec{r}(t)}{\Delta t} "$\frac{\Delta\vec{r}}{\Delta t}=\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$")
여기서 
극한을 생각하면
-\vec{r}(t)}{\Delta t} "$\frac{d\vec{r}}{dt}=\lim_{\Delta t\to 0}\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$")
그리고 =f(t)i+g(t)j+h(t)k "$\vec{r}(t)=f(t)i+g(t)j+h(t)k$")
라면
-\vec{r}(t) "$\vec{r}(t+\Delta t)-\vec{r}(t)$")
![$=[f(t+\Delta t)i+g(t+\Delta t)j+h(t+\Delta t)k]-[f(t)i+g(t)j+h(t)k]$](/123/cgi-bin/mimetex.cgi?\Large =[f(t+\Delta t)i+g(t+\Delta t)j+h(t+\Delta t)k]-[f(t)i+g(t)j+h(t)k] "$=[f(t+\Delta t)i+g(t+\Delta t)j+h(t+\Delta t)k]-[f(t)i+g(t)j+h(t)k]$")
![$=[f(t+\Delta t)-f(t)]i+[g(t+\Delta t)-g(t)]j+[h(t+\Delta t)-h(t)]k$](/123/cgi-bin/mimetex.cgi?\Large =[f(t+\Delta t)-f(t)]i+[g(t+\Delta t)-g(t)]j+[h(t+\Delta t)-h(t)]k "$=[f(t+\Delta t)-f(t)]i+[g(t+\Delta t)-g(t)]j+[h(t+\Delta t)-h(t)]k$")
따라서 구하고자 하는 식은
-\vec{r}(t)}{\Delta t} "$\lim_{\Delta t\to 0}\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$")
-f(t)}{\Delta t}\right)i+\lim_{\Delta t\to 0}(...)j+\lim_{\Delta t\to 0}(...)k "$=\lim_{\Delta t\to 0}\left(\frac{f(t+\Delta t)-f(t)}{\Delta t}\right)i+\lim_{\Delta t\to 0}(...)j+\lim_{\Delta t\to 0}(...)k$")
Leibniz 표기법:
Lagrange 표기법:
=f%27(t)\hat{\rm i}+g%27(t)\hat{\rm j}+h%27(t)\hat{\rm k} "$\vec{r}{}'(t)=f'(t)\hat{\rm i}+g'(t)\hat{\rm j}+h'(t)\hat{\rm k}$")
(https://youtu.be/40r56pX4mqA 3:30)