A basic relation in spherical coordinates

Question

Why is it that $$x\partial_x+y\partial_y+z\partial_z=r\partial_r~?$$ I know that $$r^2=x^2+y^2+z^2,$$ but how is this relation implied?

Answer 1

In Cartesian coordinates, $$ {\bf r} = x \ {\hat{\bf{ x}}} + y \ {\hat{\bf{ y}}} + z \ {\hat{\bf{ z}}} $$ and $$ \nabla = {\hat{\bf{ x}}} \ \partial_x + {\hat{\bf{ y}}} \ \partial_y + {\hat{\bf{ z}}} \ \partial_z $$

In Spherical coordinates, $$ {\bf r} = r \ {\hat{\bf{ r}}} $$ and $$ \nabla = {\hat{\bf{ r}}} \ \partial_r + {\hat{\bf{ \theta}}} \ \frac{1}{r} \partial_\theta + {\hat{\bf{ \phi}}} \ \frac{1}{r \sin \theta} \partial_\phi $$

Now evaluate ${\bf r} \bullet \nabla$ in both coordinate systems.