- #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.