position vector

In the space ${ℝ}^3$, the vector

$$\overrightarrow{r}:=(x,y,z)=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$$

directed from the origin to a point  $(x,y,z)$  is the position vector of this point. When the point is , $\overrightarrow{r}$ a vector field and its

$$r:=\sqrt{x^2+y^2+z^2}$$

a scalar .

The

• $\nabla \cdot \overrightarrow{r}=\mathrm{\hspace{0.33em}3}$

• $\nabla \times \overrightarrow{r}=\overrightarrow{0}$

• $\nabla r={\displaystyle \frac{\overrightarrow{r}}{r}}={\overrightarrow{r}}^{\mathrm{\hspace{0.17em}0}}$

• $\nabla {\displaystyle \frac{1}{r}}=-{\displaystyle \frac{\overrightarrow{r}}{r^3}}=-{\displaystyle \frac{{\overrightarrow{r}}^{\mathrm{\hspace{0.17em}0}}}{r^2}}$

• ${\nabla }^2{\displaystyle \frac{1}{r}}=\mathrm{\hspace{0.33em}0}$

are valid, where ${\overrightarrow{r}}^{\mathrm{\hspace{0.17em}0}}$ is the unit vector having the direction of $\overrightarrow{r}$.

If  $\overrightarrow{c}$  is a vector,  $\overrightarrow{U}:{ℝ}^3\to {ℝ}^3$  a vector function and  $f:ℝ\to ℝ$  is a twice differentiable function, then the formulae

• $\nabla (\overrightarrow{c}\cdot \overrightarrow{r})=\overrightarrow{c}$

• $\nabla \cdot (\overrightarrow{c}\times \overrightarrow{r})=\mathrm{\hspace{0.33em}0}$

• $(\overrightarrow{U}\cdot \nabla )\overrightarrow{r}=\overrightarrow{U}$

• $(\overrightarrow{U}\times \nabla )\cdot \overrightarrow{r}=\mathrm{\hspace{0.33em}0}$

• $(\overrightarrow{U}\times \nabla )\times \overrightarrow{r}=-2\overrightarrow{U}$

• $\nabla f(r)=f^{\prime }(r){\overrightarrow{r}}^{\mathrm{\hspace{0.17em}0}}$

• ${\nabla }^{2}f(r)=f^{\prime \prime }(r)+{\displaystyle \frac{2}{r}}{f}^{\prime }(r)$

hold.