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by
Ciro Santilli
(
@cirosantilli,
34
)
Divergence in Einstein notation
(
∂
i
)
...
Area of mathematics
Algebra
Linear algebra
Tensor
Einstein notation
Einstein notation for partial derivatives
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Updated
2024-12-15
+
Created
1970-01-01
See my version
First we write a
vector field
as:
F
(
x
0
,
x
1
,
x
2
)
=
(
F
0
(
x
0
,
x
1
,
x
2
)
,
F
1
(
x
0
,
x
1
,
x
2
)
,
F
2
(
x
0
,
x
1
,
x
2
)
)
:
R
3
→
R
3
(1)
Note how we are denoting each component of
F
as
F
i
with a
raised index
.
Then, the
divergence
can be written in
Einstein notation
as:
∇
⋅
F
=
∂
x
0
∂
F
0
(
x
0
,
x
1
,
x
2
)
+
∂
x
1
∂
F
1
(
x
0
,
x
1
,
x
2
)
+
∂
x
2
∂
F
2
(
x
0
,
x
1
,
x
2
)
=
∂
i
F
i
(
x
0
,
x
1
,
x
2
)
=
∂
x
i
∂
F
i
(
x
0
,
x
1
,
x
2
)
(2)
It is common to just omit the variables of the function, so we tend to just say:
∇
⋅
F
=
∂
i
F
i
(3)
or equivalently when referring just to the
operator
:
∇
⋅
=
∂
i
(4)
Ancestors
(8)
Einstein notation for partial derivatives
Einstein notation
Tensor
Linear algebra
Algebra
Area of mathematics
Mathematics
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Einstein notation for partial derivatives
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