In the space
, the vector
directed from the origin to a point \,
\, is the position vector of this point.\, When the point is variable,
represents a vector field and its length
a scalar field.
The simple formulae
![{\displaystyle \nabla \!\cdot {\vec {r}}=3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/934865e5f0c6c00cafa8af5fa16300b6b946aa70)
![{\displaystyle \nabla \!\times \!{\vec {r}}={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c83938b2fd68934357a2ad43409e48d11acd1ce6)
![{\displaystyle \nabla r={\frac {\vec {r}}{r}}={\vec {r}}^{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcda9217bd151d980f117421eea6e49932454120)
![{\displaystyle \nabla {\frac {1}{r}}=-{\frac {\vec {r}}{r^{3}}}=-{\frac {{\vec {r}}^{0}}{r^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1c45a39cd5746ee192c92530c66ee64b524763)
![{\displaystyle \nabla ^{2}{\frac {1}{r}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b19950916075d86a41c0e0e1128872239ff6036)
are valid, where
is the unit vector having the direction of
.
If\, Failed to parse (syntax error): {\displaystyle \vec}
\, is a constant vector,\,
\, a vector function and\,
\, is a twice differentiable function, then the formulae
- Failed to parse (syntax error): {\displaystyle \nabla(\vec\cdot\!\vec{r}) = \vec}
![{\displaystyle \nabla \cdot ({\vec {\times }}{\vec {r}})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71fe388b63ffddde634a635b0acf741a096458a9)
![{\displaystyle ({\vec {U}}\!\cdot \!\nabla ){\vec {r}}={\vec {U}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa105d0f8876828c3422c0acdde80be2d91dcde)
![{\displaystyle ({\vec {U}}\!\times \!\nabla )\!\cdot \!{\vec {r}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b507b1cb38381dd308153bb8890867696843ddf1)
![{\displaystyle ({\vec {U}}\!\times \!\nabla )\!\times \!{\vec {r}}=-2{\vec {U}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41be2e107097a7f554b11f9dc131c8704467624b)
![{\displaystyle \nabla f(r)=f'(r)\,{\vec {r}}^{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75cd9dd3ce3c178d6860651a1d4ef54d7ebc7b3e)
![{\displaystyle \nabla ^{2}f(r)=f''(r)\!+{\frac {2}{r}}f'(r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72617974b2e7fedea7ca11140e274b7d123877cc)
hold.
- ↑ {\sc K. V\"ais\"al\"a:} Vektorianalyysi . \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961).