In the space $\mathbb{R}^3$ , the vector $$\vec{r} := (x,\,y,\,z) = x\vec{i}+y\vec{j}+z\vec{k}$$ directed from the origin to a point $(x,\,y,\,z)$ is the position vector of this point. When the point is variable, $\vec{r}$ represents a vector field and its length $$r := \sqrt{x^2+y^2+z^2}$$ a scalar field.

The simple formulae

• $\nabla\!\cdot\vec{r} = 3$
• $\nabla\!\times\!\vec{r} = \vec{0}$
• $\displaystyle\nabla r = \frac{\vec{r}}{r} = \vec{r}^0$
• $\displaystyle\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3} = -\frac{\vec{r}^0}{r^2}$
• $\displaystyle\nabla^2\frac{1}{r} = 0$

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

• $\nabla(\vec{c}\cdot\!\vec{r}) = \vec{c}$
• $\nabla\cdot(\vec{c}\times\vec{r}) = 0$
• $(\vec{U}\!\cdot\!\nabla)\vec{r} = \vec{U}$
• $(\vec{U}\!\times\!\nabla)\!\cdot\!\vec{r} = 0$
• $(\vec{U}\!\times\!\nabla)\!\times\!\vec{r} = -2\vec{U}$
• $\nabla f(r) = f'(r)\,\vec{r}^0$
• $\displaystyle\nabla^2f(r) = f''(r)\!+\frac{2}{r}f'(r)$

Bibliography

1

K. V¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).