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?