Index Notation and Vector Field Manipulation: Solving Complex Problems with Ease

In summary, the conversation discusses a problem with index notation and solving for an arbitrary vector field and position vector. The solution involves using the identity \epsilon_{kij} \epsilon_{klm} = (\delta_{il}\,\delta_{jm} - \delta_{im}\,\delta_{jl}) and simplifying multiple terms to arrive at the final result. There is also a correction made for a factor of 1/4 instead of 1/2. Additionally, there is a discussion about the notation \partial_j A_i and its equivalence to \nabla A, with the conclusion that it is a second rank tensor.
  • #1
Sunshine
31
0
I've been stuck with this problem since a while.. thought I'd ask here;

[tex]\nabla \times \dfrac{\vec{A} \times \vec{r}}{2}[/tex]
solving normally isn't any problem, but I have to do it with index notation, where A is an arbitrary vector field and r is the position vector)

This is how far I can come:
(leaving out the vector-lines above A and r)
[tex]\frac{1}{4}(\nabla \times (A \times r)) = \frac{1}{4}\epsilon_{ijk}\cdot \partial_{j}(A \times r)_k = \frac{1}{4}\epsilon_{ijk}\cdot \partial_{j}(\epsilon_{klm} \cdot A_l \cdot r_m) = \frac{1}{4}\epsilon_{ijk} \epsilon_{klm}\cdot \partial_{j}(A_l \cdot r_m) = \frac{1}{4}\epsilon_{ijk} \epsilon_{klm}(r_m \cdot \partial_j A_l + A_l \cdot \partial_j r_m)[/tex]

But then what...? I'm not even sure I'm allowed to bring in that [tex]\epsilon_{klm}[/tex]. I'm very new to this notation, and don't know much more than einstein's summation convention.
 
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  • #2
Sunshine said:
I've been stuck with this problem since a while.. thought I'd ask here;

[tex]\nabla \times \dfrac{\vec{A} \times \vec{r}}{2}[/tex]
solving normally isn't any problem, but I have to do it with index notation, where A is an arbitrary vector field and r is the position vector)

This is how far I can come:
(leaving out the vector-lines above A and r)
[tex]\frac{1}{4}(\nabla \times (A \times r)) = \frac{1}{4}\epsilon_{ijk}\cdot \partial_{j}(A \times r)_k = \frac{1}{4}\epsilon_{ijk}\cdot \partial_{j}(\epsilon_{klm} \cdot A_l \cdot r_m) = \frac{1}{4}\epsilon_{ijk} \epsilon_{klm}\cdot \partial_{j}(A_l \cdot r_m) = \frac{1}{4}\epsilon_{ijk} \epsilon_{klm}(r_m \cdot \partial_j A_l + A_l \cdot \partial_j r_m)[/tex]

But then what...? I'm not even sure I'm allowed to bring in that [tex]\epsilon_{klm}[/tex]. I'm very new to this notation, and don't know much more than einstein's summation convention.
SOLUTION HINTS:

You did very well so far!
When indexing a vector result, remember to indicate the LEFT side component with the FREE index ("i" in this case). Thus, your result should have been (note the "i" subscript added on the LEFT side):
(Note: To save time, the constant fraction factor is not shown.)

[tex] 1: \ \ \ \ (\nabla \, \times \, (\vec{A} \, \times \, \vec{r}))_{\displaystyle i} \ \ = \ \ \epsilon_{ijk} \, \epsilon_{klm} \, (r_m \cdot \partial_j A_l \ + \ A_l \cdot \partial_j r_m) \ \ = \ \ \epsilon_{kij} \, \epsilon_{klm} \, (r_m \cdot \partial_j A_l \ + \ A_l \cdot \partial_j r_m)[/tex]

What to do next?? Almost always at this point, the following identity is invoked:

[tex] 2: \ \ \ \ \epsilon_{\displaystyle kij} \, \epsilon_{\displaystyle klm} \ \, = \ \, (\delta_{\displaystyle il}\,\delta_{\displaystyle jm} \, - \, \delta_{\displaystyle im}\,\delta_{\displaystyle jl}) [/tex]

Placing the above identity into Eq 1, we get:

[tex] 3: \ \ \ \ (\nabla \, \times \, (\vec{A} \, \times \, \vec{r}))_{\displaystyle i} \ \ = \ \ (\delta_{il}\,\delta_{jm} \, - \, \, \delta_{im}\,\delta_{jl}) \cdot (r_m \cdot \partial_j A_l \ + \ A_l \cdot \partial_j r_m) [/tex]

[tex] 4: \ \ \ \ \ \Longrightarrow \ \ \ (\nabla \, \times \, (\vec{A} \, \times \, \vec{r}))_{\displaystyle i} \ \ = \ \ (\delta_{il}\,\delta_{jm})\cdot(r_m \cdot \partial_j A_l) \ \, + \ \, (\delta_{il}\,\delta_{jm})\cdot(A_l \cdot \partial_j r_m) \ \, - \ \, (\delta_{im}\,\delta_{jl})\cdot(r_m \cdot \partial_j A_l) \ \, - \ \, (\delta_{im}\,\delta_{jl})\cdot(A_l \cdot \partial_j r_m) [/tex]

FIRST term on the right of Eq 4 is simplified below. Use this example to further simplify the entire expression and derive the final result.

[tex] 5: \ \ \, \ (\delta_{\displaystyle il}\,\delta_{\displaystyle jm})\cdot(r_{\displaystyle m} \cdot \partial_{\displaystyle j} A_{\displaystyle l}) \ \ = \ \ (r_{\displaystyle j} \cdot \partial_{\displaystyle j} A_{\displaystyle i}) \ \ \ \ \ \ \ \ \ \ \ \ \color{red} \textsf{(Example 1st term on right Eq 4)} [/tex]

(Remember to apply the constant fraction factor (1/2) when finished.)

~~
 
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  • #3
Where does that [itex] \frac{1}{4} [/itex] come from ?To my mind,there was supposed to be only [itex]\frac{1}{2} [/itex].

Daniel.
 
  • #4
dextercioby said:
Where does that [itex] \frac{1}{4} [/itex] come from ?To my mind,there was supposed to be only [itex]\frac{1}{2} [/itex].

Daniel.
Good observation. "Sunshine" should make that correction for the final result (see Msg #2).


~~
 
Last edited:
  • #5
Thanks xanthym... extremely good explanation! I wasn't aware of identity 2.

As for the 1/2 that became 1/4, I was thinking of the equation as [tex]\nabla \times (\dfrac{\vec{A}}{2} \times \dfrac{\vec{r}}{2}) [/tex] whick isn't the case. Thank you for noticing. :)
 
  • #6
stuck with the same problem: I know that

[tex]\partial_{\displaystyle i} A_{\displaystyle i}[/tex] means [tex]\nabla \cdot A[/tex] but is [tex]\partial_{\displaystyle j} A_{\displaystyle i}[/tex] equivalent to [tex]\nabla A[/tex]?
 
  • #7
Nope,that's a second rank tensor in cartesian coordinates.You can see it has 9 components,i.e.the components of the GRADIENT of the vector field [itex] \vec{A} [/itex].

Daniel.
 

1. What is index notation?

Index notation is a way of writing mathematical expressions using indices or powers. It is also known as exponential notation or simply as exponents.

2. How do you solve with index notation?

To solve with index notation, you must follow the rules of exponents. This includes multiplying and dividing numbers with the same base, raising a power to a power, and simplifying expressions with negative or fractional exponents.

3. What are the advantages of using index notation?

Index notation allows for easier and more efficient calculation of large or small numbers. It also simplifies expressions and makes them easier to understand and work with.

4. Can index notation be used in other fields besides mathematics?

Yes, index notation is commonly used in fields such as physics, chemistry, and engineering to represent quantities with large or small values. It is also used in computer science for data representation and algorithms.

5. What are some common mistakes to avoid when using index notation?

Some common mistakes to avoid when using index notation include forgetting to apply the rules of exponents, mixing up the order of operations, and incorrectly simplifying expressions. It is important to carefully follow the rules and double check your work to avoid errors.

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