- #1

lostminty

- 82

- 0

## Homework Statement

[itex]\frac{∂x_{i}}{∂x_{j}}[/itex] = δ[itex]_{ij}[/itex]

## Homework Equations

[itex]\vec{r}[/itex] = x[itex]_{i}[/itex]e[itex]_{i}[/itex]

## The Attempt at a Solution

[itex]\frac{∂x_{i}}{∂x_{j}}[/itex] = 1 iff i=j

δ[itex]_{ij}[/itex] = 1 iff i=j

therefore

[itex]\frac{∂x_{i}}{∂x_{j}}[/itex] = δ[itex]_{ij}[/itex]

## Homework Statement

r[itex]^{2}[/itex] = x[itex]_{k}[/itex]x[itex]_{k}[/itex]

## Homework Equations

[itex]\vec{r}[/itex] = x[itex]_{k}[/itex]e[itex]_{k}[/itex]

[itex]\vec{r}[/itex] = x[itex]_{j}[/itex]e[itex]_{j}[/itex]

## The Attempt at a Solution

r[itex]^{2}[/itex] = x[itex]_{k}[/itex]e[itex]_{k}[/itex][itex]\bullet[/itex]x[itex]_{j}[/itex]e[itex]_{j}[/itex]

= e[itex]_{k}[/itex]e[itex]_{j}[/itex]x[itex]_{k}[/itex]x[itex]_{j}[/itex]

e[itex]_{k}[/itex]e[itex]_{j}[/itex] = δ[itex]_{jk}[/itex] = 1 iff j=k

r[itex]^{2}[/itex] = x[itex]_{k}[/itex]x[itex]_{k}[/itex] iff j=k

## Homework Statement

([itex]\nabla[/itex]r[itex]^{2}[/itex])[itex]_{j}[/itex]= [itex]\frac{∂}{∂x_{j}}[/itex]([itex]x_{l}[/itex][itex]x_{l}[/itex])= 2x[itex]_{j}[/itex]

## Homework Equations

r[itex]^{2}[/itex] = [itex]x_{k}[/itex][itex]x_{k}[/itex]

## The Attempt at a Solution

pretty confused by now, so far I've guessed my way through.

is the j index communative? if so where did the l index come from. there is only 2 l's so maybe they cancel and you get x[itex]^{2}[/itex] which differentiation gives 2x. and it inherits the i index from the d/dxi.