For my continuum mechanics class, I'm tasked with finding $\nabla u$, $u$ being $u = b\frac{x}{|x|^3}$. Here, $b$ is a scalar constant. Attempt at the solution: I rewrite $\frac{1}{|x|^3}$ as $(x_i^2)^{\frac{-3}{2}}$, in order to express $u$ as a product of two functions. Now I'm at: \begin{equation} \nabla u = \frac{\delta}{\delta x_j} bx_i(x_i^2)^{\frac{-3}{2}} \end{equation} Applying the product rule then the chain rule, I end up at: \begin{equation} b[(x_i^2)^{\frac{-3}{2}}\delta_{ij}+x_i(\frac{-3}{2}(x_i^2)^{\frac{-5}{2}}\frac{\delta}{\delta x_j}(x_i^2) \end{equation}
And already here I feel like I'm running into issues where I have repeating indices, when I know conceptually and intuitively they should be free. I know what I want to do, but I'm obviously not getting the indices to work out. I'm wondering if I'm going about this the wrong way, or if maybe I'm just entirely wrong about how to take the gradient of a vector field. Any help would be appreciated!
The main problem seems to be in writing $x_i^2$ in your first line. With the summation convention you could write this as $$\frac{x_i}{\left(x_k x_k\right)^{3/2}}.$$ But the expression you have written, $\frac{x_i}{\left(x_i^2\right)^{3/2}}$, uses the same index both for the vector in the numerator and (what should be) the sum leading to a real number in the denominator, which results in the confusion you mentioned that it looks like the $x_i$ in the numerator is being summed over $i$.