Suppose that in cartesian coordinates, we have
p
→
=
(
x
0
,
y
0
,
z
0
)
{\displaystyle {\vec {p}}=(x_{0},y_{0},z_{0})}
F
→
(
p
→
)
=
f
1
(
x
0
,
y
0
,
z
0
)
i
→
+
f
2
(
x
0
,
y
0
,
z
0
)
j
→
+
f
3
(
x
0
,
y
0
,
z
0
)
k
→
{\displaystyle {\vec {F}}({\vec {p}})=f_{1}(x_{0},y_{0},z_{0}){\vec {i}}+f_{2}(x_{0},y_{0},z_{0}){\vec {j}}+f_{3}(x_{0},y_{0},z_{0}){\vec {k}}}
f s have continuous partial derivatives. Let S be a sphere of radius r centered at
p
→
{\displaystyle {\vec {p}}}
lim
r
→
0
(
1
4
3
π
r
3
∬
S
⊂
⊃
F
→
⋅
n
→
S
d
S
)
=
∂
f
1
∂
x
+
∂
f
2
∂
y
+
∂
f
3
∂
z
{\displaystyle \lim _{r\to 0}\left({\frac {1}{{\frac {4}{3}}\pi r^{3}}}\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset {\vec {F}}\cdot {\vec {n}}_{S}dS\right)={\frac {\partial f_{1}}{\partial x}}+{\frac {\partial f_{2}}{\partial y}}+{\frac {\partial f_{3}}{\partial z}}}
