Understanding Index Notation in Engineering and Physics

[chusifer]

hello, i just started learning index notation in my engineering class, and I am having some trouble. one of the problems on my homework was:

putting this in index notation:
$$\vec{f}=g \frac{m_1m_2}{\vec{r}^2} \ \frac{\vec{r}}{\sqrt{\vec{r}^2}}$$

and then another problem that reads...

consider a plane with outer normal vector $$v_i$$ on which a force is acting $$f_i$$. what are the normal components of force on the surface? wut is the max shear component? what direction is the max shear component pointing? write this in index notation.

i can figure out the components...just not how to write them in index notation. so any help here would be appreciated. thanks

 

[arildno]

For your f, you may write:
$$f_{i}=\frac{Gm_{1}m_{2}}{r_{j}^{2}}\frac{r_{i}}{\sqrt{r_{k}^{2}}}$$
I read "g" as "G" ; hope I was right about that..

 

[robphy]

$$f^a=g \frac{m_1m_2}{(r_b r^b)^{3/2}} \ r^a$$


Given a vector $\vec f$ and a unit-vector $\hat n$,
the vector component of $\vec f$ along $\hat n$ is
$(\vec f\cdot \hat n)\hat n$.
In index notation,
$( f^a g_{ab} n^b) n^c$, where $g_{ab}$ is the Euclidean metric tensor and $n^a g_{ab} n^b=1$.

 

[chusifer]

for the problem with the normal vector and external force, arent the components fsin and fcos? how would i turn those into index notation?

 

[robphy]

for the problem with the normal vector and external force, arent the components fsin and fcos? how would i turn those into index notation?

$$f^a g_{ab} n^b=\vec f\cdot\hat n =|\vec f | |\hat n|\cos\theta _\text{between f and n }=|\vec f | \cos\theta _\text{between f and n }$$

 

[chusifer]

hmmmm $$g_a_b$$ ...that sounds like the matrix $$B_i_j$$ my prof was talking about. I am unfamiliar with the term Euclidian metric tensor...but am i right in calling it a matrix?