How can I understand the Einstein summation convention for vector algebra?

[Matt1991]

Hi,

I am just starting to learn vector algebra with Grad, Div, Curl etc and have in passing come across Einstein notation which seems to make things much more concise.

The problem I have is in Finding Div(rⁿ r) where r =xi + yj + zk. The unbold r is the magnitude of r.

I have used some basic Einstein notation to make my working shorter but am stuck understanding a certain part of the notation which must be true to lead to the correct answer.

My Working:

$$\frac{\partial}{\partial x_{i}}\((r^n x_{i})$$


product rule:

$$= \ nr^{n-1} \frac{\partial r}{\partial x_{i}} x_{i}+r^n \frac{\partial x_{i}}{\partial x_{i}}$$

$$= \ nr^{n-1}\frac{x_{i}}{r}x_{i}\ = \ nr^{n} \r \ + \ 3 r^n$$


$$= (n+3)r^n$$


My problem is in understanding the step where $$\frac{ x_{i} x_{i}}{ r}$$ becomes $$r$$. For this to happen x_ix_i must be evaluated as x²+y²+z² (in spatial coordinates) which is the part I am having trouble understanding.

An explanation of this or if somebody could point me towards somewhere where I can get a simple explanation of this would be very much appreciated.

Thanks,

Matt

PS Sorry if the laTeX is bad. Its my first time using it.

 

[Matt1991]

I have made a correction to my original working that I posted. It is now hopefully correct.

Apologies,

Matt

 

[HallsofIvy]

Hi,

I am just starting to learn vector algebra with Grad, Div, Curl etc and have in passing come across Einstein notation which seems to make things much more concise.

The problem I have is in Finding Div(rⁿ r) where r =xi + yj + zk. The unbold r is the magnitude of r.

I have used some basic Einstein notation to make my working shorter but am stuck understanding a certain part of the notation which must be true to lead to the correct answer.

My Working:

$$\frac{\partial}{\partial x_{i}}\((r^n x_{i})$$


product rule:

$$= \ nr^{n-1} \frac{\partial r}{\partial x_{i}} x_{i}+r^n \frac{\partial x_{i}}{\partial x_{i}}$$

$$= \ nr^{n-1}\frac{x_{i}}{r}x_{i}\ = \ nr^{n} \r \ + \ 3 r^n$$


$$= (n+3)r^n$$


My problem is in understanding the step where $$\frac{ x_{i} x_{i}}{ r}$$ becomes $$r$$. For this to happen x_ix_i must be evaluated as x²+y²+z² (in spatial coordinates) which is the part I am having trouble understanding.

But you are using coordinates x₁, x₂, and x₃ in place of x, y, and z. By the Einstein summation convention, $x_iy_i$ means $x_1y_1+ x_2y_2+ x_3y_3$ so that $x_iy_i$ means $x_1x_1+ x_2x_2+ x_3x_3= x_1^2+ x_2^2+ x_3^2$ which is the same as $x^2+ y^2+ z^2$.

An explanation of this or if somebody could point me towards somewhere where I can get a simple explanation of this would be very much appreciated.

Thanks,

Matt

PS Sorry if the laTeX is bad. Its my first time using it.

 

[Matt1991]

Thanks for the response,

Ah right. If that is the convention then I can definitely see why. I am not sure I understand the reasoning behind the convention though.

if x_i is simply x₁ + x₂ + x₃ + ...

then I am not sure what the reasoning is behind x_i multiplied by itself acting as the einstein notation suggests it does. When I imagine this as its individual spatial coordinates (or as a operations on x₁,x₂, etc) it seems to me that it should be (x₁+x₂+...)^2

Obviously my resoning is wrong, I just can't seem to figure out where.

Thanks,

Matt

 

[Rasalhague]

if x_i is simply x₁ + x₂ + x₃ + ...

On its own, x_i doesn't stand for a sum. It's just one variable: x or y or z. The summation convention only applies when two variables in the same term have the same index:

x_ix_i = x₁x₁ + x₂x₂ + x₃x₃ = x² + y² + z².

a_ib_i = a₁b₁ + a₂b₂ + a₃b₃

(Incidentaly, if you're using this convention and you happen to have two variables with the same index in a term but don't want it to denote a sum, just write "no sum on i" or "no sum over k" or whatever the index is.)

 

[Edgardo]

You can find out more about the Einstein summation convention here:

1) See http://www.ph.ed.ac.uk/~martin/mp2h/VTF/lecture05.pdf" course.
(Martin Evans, University of Edinburgh, http://www.ph.ed.ac.uk/" )

2) http://www.luc.edu/faculty/dslavsk/courses/phys301/classnotes/einsteinsummationnotation.pdf" by David Slavsky
(Physics 301/Math 355: Mathematical Methods of Physics, Loyola University Chicago)

3) http://www.cs.caltech.edu/~cs20/c/esn-v205.pdf" by Alan H. Barr, California Institute of Technology

4) Via the http://en.wikipedia.org/wiki/Einstein_notation" in which Einstein introduces his notation (page 158 of the document or page 8 of the PDF).

5) John Armstrong explains why Einstein introduced his notation in http://unapologetic.wordpress.com/2008/05/21/the-einstein-summation-convention/" .