Divergence of the three-dimensional radial vector field $\vec{r}$ is

Radial Vector Field Divergence Calculation

This question asks us to find the divergence of the three-dimensional radial vector field, denoted as $\vec{r}$.

Understanding the Radial Vector Field

In a three-dimensional Cartesian coordinate system, the radial vector field $\vec{r}$ can be represented as:

$$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$$

where $x$, $y$, and $z$ are the coordinates, and $\hat{i}$, $\hat{j}$, $\hat{k}$ are the unit vectors along the x, y, and z axes, respectively.

Calculating Divergence

The divergence of a vector field $\vec{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$ is a scalar quantity calculated using the dot product of the del operator ($\nabla$) and the vector field:

$$\nabla \cdot \vec{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$

For the radial vector field $\vec{r}$, we have:

- $F_x=x$
- $F_y=y$
- $F_z=z$

Now, we calculate the partial derivatives:

- $$\frac{\partial F_x}{\partial x}=\frac{\partial x}{\partial x}=1$$
- $$\frac{\partial F_y}{\partial y}=\frac{\partial y}{\partial y}=1$$
- $$\frac{\partial F_z}{\partial z}=\frac{\partial z}{\partial z}=1$$

Finally, we sum these partial derivatives to find the divergence:

$$\nabla \cdot \vec{r}=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=1+1+1=3$$

Conclusion

The divergence of the three-dimensional radial vector field $\vec{r}$ is 3.