Notation - Two adjacent vectors?

I'm researching multivariable calculus presently and also have found formulas entailing 2 bolded variables positioned alongside, thus:

$$ \nabla \mathbf{f}=\frac{\partial {{f}_{j}}}{\partial {{x}_{i}}}{{\mathbf{e}}_{i}}{{\mathbf{e}}_{j}} $$

Is this suggested to be a dot item? Or another thing?

2022-06-07 14:33:54
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Answers: 1

Since the outcome of $\nabla\mathbf{f}$ offers us a 2nd - ranking tensor (it can be stood for as a 2 - by - 2 matrix with parts $\frac{\partial f^j}{\partial x^i}$), the common vector basis $\mathbf{e}_i=(0,\cdots,1,0,\cdots,0)$ (where the one remains in the $i^{th}$ area) does not fairly suffice. Nonetheless, we can generalise them. If we specify the matrices $\mathbf{e}_i\mathbf{e}_j$ to have all absolutely nos with the exception of a one in the $(i,j)$ place, after that the amount $$\sum_{i,j=1}^n \frac{\partial f^j}{\partial x^i} \mathbf{e}_i\mathbf{e}_j$$ offers us the wanted matrix, equally as the amount $\sum_{i=1}^n \frac{\partial g}{\partial x^i} \mathbf{e}_i$ offers us $\nabla g$.

Effectively, we need to be creating $\mathbf{e}_i\bigotimes\mathbf{e}_j$ as we are in fact managing the Kronecker product of $\mathbf{e}_i$ and also $\mathbf{e}_j$, though in several contexts it appears individuals go down the straight amount icon.

2022-06-07 14:52:56