In the space
R
3
{\displaystyle \mathbb {R} ^{3}}
, the vector
r
→
:=
(
x
,
y
,
z
)
=
x
i
→
+
y
j
→
+
z
k
→
{\displaystyle {\vec {r}}\;:=\;(x,\,y,\,z)\;=\;x{\vec {i}}+y{\vec {j}}+z{\vec {k}}}
directed from the origin to a point \,
(
x
,
y
,
z
)
{\displaystyle (x,\,y,\,z)}
\, is the position vector of this point.\, When the point is variable,
r
→
{\displaystyle {\vec {r}}}
represents a vector field and its length
r
:=
x
2
+
y
2
+
z
2
{\displaystyle r\;:=\;{\sqrt {x^{2}+y^{2}+z^{2}}}}
a scalar field.
The simple formulae
∇
⋅
r
→
=
3
{\displaystyle \nabla \!\cdot {\vec {r}}=3}
∇
×
r
→
=
0
→
{\displaystyle \nabla \!\times \!{\vec {r}}={\vec {0}}}
∇
r
=
r
→
r
=
r
→
0
{\displaystyle \nabla r={\frac {\vec {r}}{r}}={\vec {r}}^{0}}
∇
1
r
=
−
r
→
r
3
=
−
r
→
0
r
2
{\displaystyle \nabla {\frac {1}{r}}=-{\frac {\vec {r}}{r^{3}}}=-{\frac {{\vec {r}}^{0}}{r^{2}}}}
∇
2
1
r
=
0
{\displaystyle \nabla ^{2}{\frac {1}{r}}=0}
are valid, where
r
→
0
{\displaystyle {\vec {r}}^{0}}
is the unit vector having the direction of
r
→
{\displaystyle {\vec {r}}}
.
If\, Failed to parse (syntax error): {\displaystyle \vec}
\, is a constant vector,\,
U
→
:
R
3
→
R
3
{\displaystyle {\vec {U}}\!\!:\mathbb {R} ^{3}\to \mathbb {R} ^{3}}
\, a vector function and\,
f
:
R
→
R
{\displaystyle f\!\!:\mathbb {R} \to \mathbb {R} }
\, is a twice differentiable function , then the formulae
Failed to parse (syntax error): {\displaystyle \nabla(\vec\cdot\!\vec{r}) = \vec}
∇
⋅
(
×
→
r
→
)
=
0
{\displaystyle \nabla \cdot ({\vec {\times }}{\vec {r}})=0}
(
U
→
⋅
∇
)
r
→
=
U
→
{\displaystyle ({\vec {U}}\!\cdot \!\nabla ){\vec {r}}={\vec {U}}}
(
U
→
×
∇
)
⋅
r
→
=
0
{\displaystyle ({\vec {U}}\!\times \!\nabla )\!\cdot \!{\vec {r}}=0}
(
U
→
×
∇
)
×
r
→
=
−
2
U
→
{\displaystyle ({\vec {U}}\!\times \!\nabla )\!\times \!{\vec {r}}=-2{\vec {U}}}
∇
f
(
r
)
=
f
′
(
r
)
r
→
0
{\displaystyle \nabla f(r)=f'(r)\,{\vec {r}}^{0}}
∇
2
f
(
r
)
=
f
″
(
r
)
+
2
r
f
′
(
r
)
{\displaystyle \nabla ^{2}f(r)=f''(r)\!+{\frac {2}{r}}f'(r)}
hold.
[ 1]
↑ {\sc K. V\"ais\"al\"a:} Vektorianalyysi . \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961).