Suffix Notation Help: Nabla Vector Calculation

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In summary, the equation for homework statement is:\nabla \times \vec{p} = -\frac{\vec{B}}{r^3} + 3\frac{\vec{B} \bullet \vec{r}}{r^5}\vec{r}
  • #1
Matt atkinson
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Homework Statement


Show that the equation [tex] \nabla \times \vec{p} = -\frac{\vec{B}}{r^3} + 3\frac{\vec{B} \bullet \vec{r}}{r^5}\vec{r} [/tex]
Where ;
[tex]\vec{p} = \frac{\vec{B} \times \vec{r}}{r^3} [/tex]
[tex] \vec{r}=(x_1 ,x_2 ,x_3) [/tex]
and [tex] \vec{B}[/tex] is a constant vector.
and r is the magnitude of [tex] \vec{r} [/tex]

Homework Equations


above

The Attempt at a Solution


[tex] \nabla \times \vec{p} = \epsilon_{ijk} \epsilon_{klm} \frac{d}{dx_j} B x_m |r|^{-3} [/tex]
[tex] = \epsilon_{ijk} \epsilon_{klj} B |r|^{-3} - 3 \epsilon_{ijk} \epsilon_{klm} B x_m x_j |r|^{-5} [/tex]
I've tried expanding and using various identities such as;
[tex] \epsilon_{ijk} \epsilon_{klm} = \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} [/tex]
If someone could give me a push in the right direction or let me know if i went wrong somewhere (i know i skipped a few steps but it took me 20 mins to right out the latex code for this adk if you don't see what I did).
 
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  • #2
Matt atkinson said:

Homework Statement


Show that the equation [tex] \nabla \times \vec{p} = -\frac{\vec{B}}{r^3} + 3\frac{\vec{B} \bullet \vec{r}}{r^5}\vec{r} [/tex]
Where ;
[tex]\vec{p} = \frac{\vec{B} \times \vec{r}}{r^3} [/tex]
[tex] \vec{r}=(x_1 ,x_2 ,x_3) [/tex]
and [tex] \vec{B}[/tex] is a constant vector.
and r is the magnitude of [tex] \vec{r} [/tex]

Homework Equations


above

The Attempt at a Solution


[tex] \nabla \times \vec{p} = \epsilon_{ijk} \epsilon_{klm} \frac{d}{dx_j} B x_m |r|^{-3} [/tex]

You're missing an [itex]l[/itex] suffix on the [itex]B[/itex] and the derivative is partial:
[tex]
(\nabla \times \vec p)_i = \epsilon_{ijk} \epsilon_{klm} B_l \frac{\partial}{\partial x_j} \left(\frac{x_m}{r^3}\right)[/tex]

You will also need
[tex]
\frac{\partial r}{\partial x_j} = \frac{x_j}{r}[/tex] and [tex]
\frac{\partial x_m}{\partial x_j} = \delta_{jm}
[/tex]
 
  • #3
ah okay, hmm.
so I should use the product rule and get
[tex] \epsilon_{ijk} \epsilon_{klm} B_l x_m \frac{\partial r^{-3}}{\partial x_j} + r^{-3} \delta_{jm} [/tex]
I just don't see how to use [tex]\frac{\partial r}{\partial x_j}[/tex] maybe I'm being a little stupid.
 
Last edited by a moderator:
  • #4
Matt atkinson said:
ah okay, hmm.
so I should use the product rule and get
[tex] \epsilon_{ijk} \epsilon_{klm} B_l x_m \frac{\partial r^{-3}}{\partial x_j} + r^{-3} \delta_{jm} [/tex]
You are missing some brackets: you should have
[tex]
\epsilon_{ijk} \epsilon_{klm} B_l \left(x_m \frac{\partial r^{-3}}{\partial x_j} + r^{-3} \delta_{jm}\right)
[/tex]


I just don't see how to use [tex]\frac{\partial r}{\partial x_j}[/tex] maybe I'm being a little stupid.

[tex]
\frac{\partial}{\partial x_j} \left(\frac{1}{r^3}\right) = \frac{d (r^{-3})}{dr} \frac{\partial r}{\partial x_j}
[/tex]
 
  • #5
Thankyou so much, I've been trying to do this for a while now, can't believe it was chain rule i forgot
 
  • #6
pasmith said:
You are missing some brackets: you should have
[tex]
\epsilon_{ijk} \epsilon_{klm} B_l \left(x_m \frac{\partial r^{-3}}{\partial x_j} + r^{-3} \delta_{jm}\right)
[/tex]

[tex]
\frac{\partial}{\partial x_j} \left(\frac{1}{r^3}\right) = \frac{d (r^{-3})}{dr} \frac{\partial r}{\partial x_j}
[/tex]

Okay, I am sorry i got stuck again, so i did the math and substitute the two epsilons for deltas uisng the identities i gave before, but i can't seem to cancel;
[tex] \delta_{il}\delta_{jj} B_jr^{-3} -\delta_{ij}\delta_{jl} B_jr^{-3} -3\delta_{il}\delta_{jm}B_j x_m x_j r^{-5}+3\delta_{im}\delta_{jl}B_j x_m x_j r^{-5} [/tex]
to the show that answer I am assuming the first and third term are zero? but I am not sure why, sorry about all the questions this suffix notation is very new to me.
 
Last edited:
  • #7
Matt atkinson said:
Okay, I am sorry i got stuck again, so i did the math and substituent the two epsilons for deltas uisng the identities i gave before, but i can't seem to cancel;
[tex] \delta_{il}\delta_{jj} B_jr^{-3} -\delta_{ij}\delta_{jl} B_jr^{-3} -3\delta_{il}\delta_{jm}B_j x_m x_j r^{-5}+3\delta_{im}\delta_{jl}B_j x_m x_j r^{-5} [/tex]
The suffix on [itex]B[/itex] should be an [itex]l[/itex]; try again with
[tex]
\delta_{il}\delta_{jj} B_l r^{-3} -\delta_{ij}\delta_{jl} B_l r^{-3} -3\delta_{il}\delta_{jm}B_l x_m x_j r^{-5}+3\delta_{im}\delta_{jl}B_l x_m x_j r^{-5}[/tex]
 
  • #8
pasmith said:
The suffix on [itex]B[/itex] should be an [itex]l[/itex]; try again with
[tex]
\delta_{il}\delta_{jj} B_l r^{-3} -\delta_{ij}\delta_{jl} B_l r^{-3} -3\delta_{il}\delta_{jm}B_l x_m x_j r^{-5}+3\delta_{im}\delta_{jl}B_l x_m x_j r^{-5}[/tex]

okay, I did that and got;
[tex] 3B_i r^{-3} - B_i r^{-3} -3B_i x_j x_j r^{-5}+3B_j x_i x_j r^{-5} [/tex]
I just don't see how two of the terms cancel, not sure what I am missing
 
Last edited:
  • #9
Matt atkinson said:
okay, I did that and got;
[tex] 3B_i r^{-3} - B_i r^{-3} -3B_i x_j x_j r^{-5}+3B_j x_i x_j r^{-5} [/tex]
I just don't see how two of the terms cancel, not sure what I am missing

[itex]x_jx_j = r^2[/itex].
 
  • #10
pasmith said:
[itex]x_jx_j = r^2[/itex].

Wow thanks I feel kinda silly now.
Thanks so much for your help.
 

FAQ: Suffix Notation Help: Nabla Vector Calculation

What is suffix notation in the context of nabla vector calculation?

Suffix notation is a mathematical notation used to express vector and tensor equations in a concise and efficient way. In nabla vector calculation, it is a way to represent the gradient, divergence, and curl operations using suffixes to denote the different components of a vector or tensor.

How does suffix notation simplify nabla vector calculations?

Suffix notation simplifies nabla vector calculations by providing a compact way to express complex vector and tensor equations. It eliminates the need for writing out each component of a vector or tensor, making calculations faster and more efficient.

What are the basic rules for using suffix notation in nabla vector calculations?

There are three basic rules for using suffix notation in nabla vector calculations: (1) repeated indices in a term are summed over, (2) an index that appears once as a superscript and once as a subscript is called a "dummy" index and does not affect the value of the term, and (3) an index that appears twice, once as a superscript and once as a subscript, is called a "free" index and must take on all possible values.

Can suffix notation be extended to higher dimensions in nabla vector calculations?

Yes, suffix notation can be extended to higher dimensions in nabla vector calculations. In three dimensions, three indices are used to represent the components of a vector or tensor (x, y, and z). In higher dimensions, the number of indices increases accordingly.

Are there any limitations to using suffix notation in nabla vector calculations?

One limitation of suffix notation in nabla vector calculations is that it can become cumbersome and difficult to read when dealing with complex equations involving multiple vectors and tensors. In these cases, other notations, such as index notation or matrix notation, may be more suitable.

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