1. ## Elemental displacement question

I was reading though my lecture notes and elemental displacement is given as $\displaystyle \delta{\underline{\textbf{r}}} = \underline{\textbf{r}}_b - \underline{\textbf{r}}_a$ for two points $\displaystyle \underline{\textbf{r}}_a = x_1\underline{\textbf{i}} + x_2\underline{\textbf{j}} + x_3\underline{\textbf{z}}$ and $\displaystyle \underline{\textbf{r}}_b = (x_1 + \delta{x_1})\underline{\textbf{i}} + (x_2 + \delta{x_2})\underline{\textbf{j}} + (x_3 + \delta{x_3})\underline{\textbf{z}}$

but then elemental distance is given as $\displaystyle \delta{\underline{\textbf{s}}^2} = \delta{\underline{\textbf{r}}}.\delta{\underline{\ textbf{r}}}$ and i dont understand how this could be true as the square of displacement is not the distance... maybe theres something im missing something

2. But what are $\displaystyle \delta{\underline{\textbf{r}}_b}$ and $\displaystyle \delta{\underline{\textbf{r}}_a}$?

3. oh woops its supposed to be $\displaystyle \delta{\underline{\textbf{s}}^2} = \delta{\underline{\textbf{r}}}.\delta{\underline{\ textbf{r}}}$ ive edited it now

4. Since $\displaystyle \delta{\underline{\textbf{r}}} = (\delta x_1,\delta x_2, \delta x_3)$, the dot product $\displaystyle \delta{\underline{\textbf{r}}}\cdot \delta{\underline{\textbf{r}}} = (\delta x_1)^2+(\delta x_2)^2+(\delta x_3)^2$.

5. Originally Posted by renlok
I was reading though my lecture notes and elemental displacement is given as $\displaystyle \delta{\underline{\textbf{r}}} = \underline{\textbf{r}}_b - \underline{\textbf{r}}_a$ for two points $\displaystyle \underline{\textbf{r}}_a = x_1\underline{\textbf{i}} + x_2\underline{\textbf{j}} + x_3\underline{\textbf{z}}$
Your basis vectors should be $\displaystyle \underline{\textbf{i}}$, $\displaystyle \underline{\textbf{j}}$, and $\displaystyle \underline{\textbf{k}}$, NOT $\displaystyle \underline{\textbf{z}}$

and $\displaystyle \underline{\textbf{r}}_b = (x_1 + \delta{x_1})\underline{\textbf{i}} + (x_2 + \delta{x_2})\underline{\textbf{j}} + (x_3 + \delta{x_3})\underline{\textbf{z}}$

but then elemental distance is given as $\displaystyle \delta{\underline{\textbf{s}}^2} = \delta{\underline{\textbf{r}}}.\delta{\underline{\ textbf{r}}}$ and i dont understand how this could be true as the square of displacement is not the distance... maybe theres something im missing something
That isn't distance, it is "distance squared".