Does the equation make sense if $\mathbf{v}(x,y,z)$ is not differentiable at point $(x_0,y_0,z_0)$?

Question

For a vector field $\mathbf{v}(x,y,z)$, if $\dfrac{\partial v_x}{\partial x}$ ,$\dfrac{\partial v_y}{\partial y}$, $\dfrac{\partial v_z}{\partial z}$ exist at point $(x_0,y_0,z_0)$ no matter how we rotate our Cartesian coordinate system, but $\mathbf{v}(x,y,z)$ is not differentiable at point $(x_0,y_0,z_0)$; then does the equation:

$$\nabla \cdot \mathbf{v}=f(x,y,z)$$

at $(x_0,y_0,z_0)$ make sense?

Answer 1

Note that a non-differentiable multivariable function can still have defined partial derivatives. If $\frac{\partial v_x}{\partial x}$, $\frac{\partial v_y}{\partial y}$ and $\frac{\partial v_z}{\partial z}$ are defined, then the equation makes sense. Though, if it means anything to you, the partial derivatives won't be continuous.