1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Partial derivative of inner product in Einstein Notation

  1. Feb 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Can someone please check my working, as I am new to Einstein notation:
    Calculate $$\partial^\mu x^2.$$
    2. Relevant equations
    3. The attempt at a solution

    \begin{align*}
    \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\
    &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \ \text{(by product rule and relabelling indices)} \\
    &=x^a\delta_\mu^a + x_b\delta_\mu^b \\
    &=2x_\mu.
    \end{align*}
    I'm not sure is the expression in the second term of the second line is correct, as the partial is with respect to the covariant vector but the argument is a contravariant vector.
     
  2. jcsd
  3. Feb 4, 2017 #2
    A useful tip to keep in mind is that indices keep their position (upper or lower) unless they are raised or lowered by a metric. So the equality that you wrote,
    [tex]x^a\partial^\mu x_a + x_b\partial^\mu x^b =x^a\delta_\mu^a + x_b\delta_\mu^b,[/tex] is clearly not correct.

    Firstly, ##\partial^{\mu} x_{a} = \delta_{a}^{\mu}## (notice that ##\mu## stays upper and ##a## stays lower).

    Next, you cannot simply perform the derivative ##\partial^{\mu} x^{b}## directly because the derivative is with respect to the contravariant component. So the correct way to go about doing it is to rewrite ##x^{b} = g^{cb} x_{c}##, so that ##\partial^{\mu} x^{b} = g^{cb} \partial^{\mu} x_{c} +x_{c} \partial^{\mu}g^{cb} ##. In a cartesian coordinate system, the metric is just identity and so we get the same result as you do.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Partial derivative of inner product in Einstein Notation
Loading...