Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by:

$$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial x_i\partial x_j} x_ix_j.$$ Or if we use $b_{ij}$ for the Hessian : $$U(x_i)=\frac{1}{2}\sum_{i,j}b_{ij}x_ix_j.$$

I want to calculate the force: $k_i=\frac{-\partial U}{\partial x_i}$. This should be $-\sum_{j} b_{ji}x_j$.

My questions:

  1. Why is the summation over the index $i$ gone?

  2. Why are the indices of the hessian $b$ switched?

I hope someone can give me an answer. I put this in the physics section because it's a physics problem, but my question is actually purely mathematical.

share|cite|improve this question
up vote 4 down vote accepted

1.) The differentiation operator acting will give rise to Kronecker-Deltas since

$\frac{\partial x_a}{\partial x_b}=\delta_{ab}$ This will kill one summation.

More specifially: $\frac{\partial U}{\partial x_a}=-1/2 \sum_{ij}b_{ij}(\delta_{ai}x_j+\delta_{aj}x_i)=-1/2( \sum_{j}b_{aj}x_j+\sum_{i}b_{ia}x_i)=-\sum_{j}b_{aj}x_j$. Rename j to be i and you're done.

2.) The Hessian matrix is a matrix of second-order partial derivatives; hence it is symmetric w.r.t. its indices, i.e. $b_{ij}=b_{ji}$.

share|cite|improve this answer
Thanks a lot! Really appreciate your effort :) – ComputerSaysNo Feb 18 '13 at 22:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.