The vector gradient is a mathematical operator that, analogous to the gradient of scalar quantities, describes the change in a vector-valued quantity as a function of location. The application of the vector gradient to a vector field results in a second order tensor at every location , so the result can be written as a matrix.
The vector gradient is defined for Euclidean vector spaces with a standard scalar product ( Frobenius scalar product ). The generalization to standardized spaces is called the Fréchet derivation .
definition
A vector field is a mapping that assigns each location in a vector . Here and are each Euclidean vector spaces with the standard scalar product “·”. The vector gradient applied to the vector field is defined as
F.
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:
R.
n
→
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{\ displaystyle {\ vec {F}} \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}
R.
n
{\ displaystyle \ mathbb {R} ^ {n}}
R.
m
{\ displaystyle \ mathbb {R} ^ {m}}
R.
n
{\ displaystyle \ mathbb {R} ^ {n}}
R.
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{\ displaystyle \ mathbb {R} ^ {m}}
Degree
{\ displaystyle \ operatorname {grad}}
F.
→
{\ displaystyle {\ vec {F}}}
Degree
F.
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(
∇
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F.
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)
⊤
=
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∈
R.
m
⊗
R.
n
{\ displaystyle \ operatorname {grad} {\ vec {F}} = ({\ vec {\ nabla}} \ otimes {\ vec {F}}) ^ {\ top} = ({\ vec {\ nabla}} {\ vec {F}}) ^ {\ top} \ in \ mathbb {R} ^ {m} \ otimes \ mathbb {R} ^ {n}}
It is the nabla operator , and the link " " the tensor ( outer product ). The superscript " " stands for the transposition and the space contains all tensors of the second level, the vectors from the linear map. The vector gradient is therefore the transposed dyadic product “ ” of the Nabla operator and a vector field.
∇
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∂
∂
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1
,
...
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∂
∂
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{\ displaystyle \ textstyle {\ vec {\ nabla}} = \ left ({\ frac {\ partial} {\ partial x_ {1}}}, \ ldots, {\ frac {\ partial} {\ partial x_ {n }}} \ right)}
⊗
{\ displaystyle \ otimes}
⊤
{\ displaystyle \ scriptstyle \ top}
R.
m
⊗
R.
n
{\ displaystyle \ mathbb {R} ^ {m} \ otimes \ mathbb {R} ^ {n}}
R.
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{\ displaystyle \ mathbb {R} ^ {n}}
R.
m
{\ displaystyle \ mathbb {R} ^ {m}}
⊗
{\ displaystyle \ otimes}
The directional derivative in the direction of a vector can be calculated with the vector gradient :
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∈
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{\ displaystyle {\ vec {h}} \ in \ mathbb {R} ^ {n}}
(
H
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∇
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H
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Degree
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{\ displaystyle ({\ vec {h}} \ cdot {\ vec {\ nabla}}) {\ vec {F}} = {\ vec {h}} \ cdot ({\ vec {\ nabla}} \ otimes {\ vec {F}}) = ({\ vec {\ nabla}} \ otimes {\ vec {F}}) ^ {\ top} \ cdot {\ vec {h}} = \ operatorname {grad} ({ \ vec {F}}) \ cdot {\ vec {h}} \ ,.}
In fluid mechanics , the left representation with the Nabla operator is preferred over the right, which is common in continuum mechanics . The directional derivative calculated with the help of the vector gradient corresponds to the directional derivative obtained by calculating the limit value:
(
Degree
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⋅
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d
d
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|
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lim
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s
for all
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{\ displaystyle (\ operatorname {grad} {\ vec {F}}) \ cdot {\ vec {h}} = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} s}} { \ vec {F}} ({\ vec {x}} + s {\ vec {h}}) \ right | _ {s = 0} = \ lim _ {s \ rightarrow 0} {\ frac {{\ vec {F}} ({\ vec {x}} + s {\ vec {h}}) - {\ vec {F}} ({\ vec {x}})} {s}} \ quad {\ text { for all}} \ quad \; {\ vec {x}}, {\ vec {h}} \ in \ mathbb {R} ^ {n} \ ,.}
Be the component-wise representations
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{\ displaystyle {\ vec {x}} = \ sum _ {j = 1} ^ {n} x_ {j} {\ vec {a}} _ {j} \ quad {\ text {and}} \ quad { \ vec {F}} ({\ vec {x}}) = \ sum _ {i = 1} ^ {m} F_ {i} ({\ vec {x}}) {\ vec {b}} _ { i}}
given with respect to a fixed orthonormal basis des and des . Then the gradient is calculated according to
{
a
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{\ displaystyle \ {{\ vec {a}} _ {j} \}}
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{\ displaystyle \ mathbb {R} ^ {n}}
{
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{\ displaystyle \ {{\ vec {b}} _ {i} \}}
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{\ displaystyle \ mathbb {R} ^ {m}}
Degree
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{\ displaystyle \ operatorname {grad} {\ vec {F}} = \ sum _ {j = 1} ^ {n} {\ frac {\ mathrm {d} {\ vec {F}}} {\ mathrm {d } x_ {j}}} \ otimes {\ vec {a}} _ {j} = \ sum _ {i = 1} ^ {m} \ sum _ {j = 1} ^ {n} {\ frac {\ mathrm {d} F_ {i}} {\ mathrm {d} x_ {j}}} {\ vec {b}} _ {i} \ otimes {\ vec {a}} _ {j}}
The components of this tensor agree with those of the Jacobi matrix :
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{\ displaystyle {\ vec {b}} _ {k} \ cdot (\ operatorname {grad} {\ vec {F}}) \ cdot {\ vec {a}} _ {l} = {\ frac {\ partial F_ {k}} {\ partial x_ {l}}} = (J _ {\ vec {F}}) _ {kl} \ ,.}
The vector gradient is u. a. used in continuum mechanics (e.g. in the Navier-Stokes equations ).
Occasionally, it is also defined in the literature .
Degree
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: =
∇
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F.
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{\ displaystyle \ operatorname {grad} {\ vec {F}}: = {\ vec {\ nabla}} \ otimes {\ vec {F}}}
Total differential
Consider an infinitesimal displacement for a vector field:
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{\ displaystyle {\ vec {F}} ({\ vec {r}} + \ mathrm {d} {\ vec {r}}) = {\ vec {F}} ({\ vec {r}}) + J _ {\ vec {F}} \ cdot \ mathrm {d} {\ vec {r}} = {\ vec {F}} ({\ vec {r}}) + (\ operatorname {grad} {\ vec { F}}) \ cdot \ mathrm {d} {\ vec {r}} = {\ vec {F}} ({\ vec {r}}) + (\ mathrm {d} {\ vec {r}} \ cdot \ nabla) {\ vec {F}} = {\ vec {F}} ({\ vec {r}}) + \ mathrm {d} {\ vec {F}}}
The complete or total differential of a vector field is:
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{\ displaystyle {\ vec {F}} ({\ vec {r}})}
d
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Degree
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{\ displaystyle \ mathrm {d} {\ vec {F}} = (\ operatorname {grad} {\ vec {F}}) \ cdot \ mathrm {d} {\ vec {r}}}
or in index notation
d
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∂
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{\ displaystyle \ mathrm {d} F_ {i} = \ sum _ {j} {\ frac {\ partial F_ {i}} {\ partial x_ {j}}} \ mathrm {d} x_ {j}}
The total differential of a scalar field and a vector field thus (formally) have the same form. For the total differential of a scalar field, the gradient is multiplied by the scalar differential. In the case of the total differential of a vector field, the multiplication between the gradient (matrix form) and the differential vector must be carried out as a matrix-vector product .
properties
The calculation rules are those of the Jacobi matrix. denotes the vector gradient here.
Degree
A.
→
{\ displaystyle \ operatorname {grad} {\ vec {A}}}
The following applies to all constants and vector fields :
c
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{\ displaystyle c \ in \ mathbb {R}}
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{\ displaystyle {\ vec {A}}, \, {\ vec {B}} \ colon \ mathbb {R} ^ {n} \ rightarrow \ mathbb {R} ^ {m}}
Linearity
Degree
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c
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=
c
⋅
Degree
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{\ displaystyle \ operatorname {grad} (c \ cdot {\ vec {A}}) = c \ cdot \ operatorname {grad} {\ vec {A}}}
Degree
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{\ displaystyle \ operatorname {degree} ({\ vec {A}} + {\ vec {B}}) = \ operatorname {degree} {\ vec {A}} + \ operatorname {degree} {\ vec {B} }}
Product rule
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{\ displaystyle ({\ vec {A}} \ cdot \ nabla) {\ vec {B}} = (\ operatorname {grad} {\ vec {B}}) \ cdot {\ vec {A}}}
Degree
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T
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{\ displaystyle \ operatorname {grad} ({\ vec {A}} \ cdot {\ vec {B}}) = (\ operatorname {grad} {\ vec {A}}) ^ {T} \ cdot {\ vec {B}} + (\ operatorname {grad} {\ vec {B}}) ^ {T} \ cdot {\ vec {A}}}
Degree
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)
=
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(
Degree
A.
→
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T
⋅
A.
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{\ displaystyle \ operatorname {grad} ({\ vec {A}} ^ {\, 2}) = 2 \, (\ operatorname {grad} {\ vec {A}}) ^ {T} \ cdot {\ vec {A}}}
The above relationship can be transformed especially for vector fields :
A.
→
,
B.
→
:
R.
3
→
R.
3
{\ displaystyle {\ vec {A}}, \, {\ vec {B}}: \ mathbb {R} ^ {3} \ rightarrow \ mathbb {R} ^ {3}}
Degree
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×
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+
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{\ displaystyle \ operatorname {grad} ({\ vec {A}} \ cdot {\ vec {B}}) = ({\ vec {B}} \ cdot \ nabla) {\ vec {A}} + {\ vec {B}} \ times (\ nabla \ times {\ vec {A}}) + ({\ vec {A}} \ cdot \ nabla) {\ vec {B}} + {\ vec {A}} \ times (\ nabla \ times {\ vec {B}})}
Degree
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=
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2
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{\ displaystyle \ operatorname {grad} ({\ vec {A}} ^ {\, 2}) = 2 ({\ vec {A}} \ cdot \ nabla) {\ vec {A}} + 2 {\ vec {A}} \ times (\ nabla \ times {\ vec {A}})}
Examples
Degree
r
→
=
I.
,
{\ displaystyle \ operatorname {grad} {\ vec {r}} = I,}
where is the identity matrix .
I.
{\ displaystyle I}
(
Degree
r
→
r
3
)
T
=
∇
⊗
r
→
r
3
=
(
∇
1
r
3
)
⊗
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1
r
3
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∇
⊗
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)
=
-
3
r
5
r
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⊗
r
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+
1
r
3
I.
=
-
1
r
5
(
3
r
→
⊗
r
→
-
r
2
I.
)
{\ displaystyle \ left (\ operatorname {grad} {\ frac {\ vec {r}} {r ^ {3}}} \ right) ^ {T} = \ nabla \ otimes {\ frac {\ vec {r} } {r ^ {3}}} = \ left (\ nabla {\ frac {1} {r ^ {3}}} \ right) \ otimes {\ vec {r}} + {\ frac {1} {r ^ {3}}} (\ nabla \ otimes {\ vec {r}}) = - {\ frac {3} {r ^ {5}}} {\ vec {r}} \ otimes {\ vec {r} } + {\ frac {1} {r ^ {3}}} I = - {\ frac {1} {r ^ {5}}} (3 {\ vec {r}} \ otimes {\ vec {r} } -r ^ {2} I)}
The last two formulas are e.g. B. used in Cartesian multipole expansion .
literature
Hugo Sirk: Introduction to vector calculation: For natural scientists, chemists and engineers . Springer-Verlag, 2013, ISBN 3-642-72313-6 , chap. 5.4 "The vector field and the vector gradient".
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