The divergence of a vector field is a measure of how much the vector field spreads out from a point. It is defined as the sum of the partial derivatives of the field’s components with respect to the coordinates.
The divergence of the three-dimensional radial vector field $\overrightarrow {\text{r}}$ is $\frac{3}{{\text{r}}}$. This is because the radial vector field points in the direction of increasing radius, and the divergence measures how much the field spreads out from a point. The larger the radius, the more the field spreads out, and so the larger the divergence.
Option A is incorrect because it is the value of the divergence at the origin. The divergence is not constant, so it does not have a single value.
Option B is incorrect because it is the value of the gradient of the radial vector field. The gradient is a vector field that points in the direction of greatest increase of the field, and its magnitude is the magnitude of the field’s rate of change. The radial vector field does not have a direction of greatest increase, so its gradient is zero.
Option C is incorrect because it is the unit vector in the radial direction. The unit vector in the radial direction is $\hat{\text{r}}$, but the divergence is not a vector.
Option D is incorrect because it is the triple product of $\hat{\text{i}}$, $\hat{\text{j}}$, and $\hat{\text{k}}$. The triple product is a scalar quantity, but the divergence is a vector quantity.