Vector gradient

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 ${\displaystyle {\vec {F}}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle \operatorname {grad} }$ ${\displaystyle {\vec {F}}}$

${\displaystyle \operatorname {grad} {\vec {F}}=({\vec {\nabla }}\otimes {\vec {F}})^{\top }=({\vec {\nabla }}{\vec {F}})^{\top }\in \mathbb {R} ^{m}\otimes \mathbb {R} ^{n}}$

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. ${\displaystyle \textstyle {\vec {\nabla }}=\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right)}$ ${\displaystyle \otimes }$ ${\displaystyle \scriptstyle \top }$ ${\displaystyle \mathbb {R} ^{m}\otimes \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle \otimes }$

The directional derivative in the direction of a vector can be calculated with the vector gradient : ${\displaystyle {\vec {h}}\in \mathbb {R} ^{n}}$

${\displaystyle ({\vec {h}}\cdot {\vec {\nabla }}){\vec {F}}={\vec {h}}\cdot ({\vec {\nabla }}\otimes {\vec {F}})=({\vec {\nabla }}\otimes {\vec {F}})^{\top }\cdot {\vec {h}}=\operatorname {grad} ({\vec {F}})\cdot {\vec {h}}\,.}$

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:

${\displaystyle (\operatorname {grad} {\vec {F}})\cdot {\vec {h}}=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}{\vec {F}}({\vec {x}}+s{\vec {h}})\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {{\vec {F}}({\vec {x}}+s{\vec {h}})-{\vec {F}}({\vec {x}})}{s}}\quad {\text{für alle}}\quad \;{\vec {x}},{\vec {h}}\in \mathbb {R} ^{n}\,.}$

Be the component-wise representations

${\displaystyle {\vec {x}}=\sum _{j=1}^{n}x_{j}{\vec {a}}_{j}\quad {\text{und}}\quad {\vec {F}}({\vec {x}})=\sum _{i=1}^{m}F_{i}({\vec {x}}){\vec {b}}_{i}}$

given with respect to a fixed orthonormal basis des and des . Then the gradient is calculated according to ${\displaystyle \{{\vec {a}}_{j}\}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \{{\vec {b}}_{i}\}}$ ${\displaystyle \mathbb {R} ^{m}}$

${\displaystyle \operatorname {grad} {\vec {F}}=\sum _{j=1}^{n}{\frac {\mathrm {d} {\vec {F}}}{\mathrm {d} x_{j}}}\otimes {\vec {a}}_{j}=\sum _{i=1}^{m}\sum _{j=1}^{n}{\frac {\mathrm {d} F_{i}}{\mathrm {d} x_{j}}}{\vec {b}}_{i}\otimes {\vec {a}}_{j}}$

The components of this tensor agree with those of the Jacobi matrix :

${\displaystyle {\vec {b}}_{k}\cdot (\operatorname {grad} {\vec {F}})\cdot {\vec {a}}_{l}={\frac {\partial F_{k}}{\partial x_{l}}}=(J_{\vec {F}})_{kl}\,.}$

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 . ${\displaystyle \operatorname {grad} {\vec {F}}:={\vec {\nabla }}\otimes {\vec {F}}}$

Total differential

Consider an infinitesimal displacement for a vector field:

${\displaystyle {\vec {F}}({\vec {r}}+\mathrm {d} {\vec {r}})={\vec {F}}({\vec {r}})+J_{\vec {F}}\cdot \mathrm {d} {\vec {r}}={\vec {F}}({\vec {r}})+(\operatorname {grad} {\vec {F}})\cdot \mathrm {d} {\vec {r}}={\vec {F}}({\vec {r}})+(\mathrm {d} {\vec {r}}\cdot \nabla ){\vec {F}}={\vec {F}}({\vec {r}})+\mathrm {d} {\vec {F}}}$

The complete or total differential of a vector field is: ${\displaystyle {\vec {F}}({\vec {r}})}$

${\displaystyle \mathrm {d} {\vec {F}}=(\operatorname {grad} {\vec {F}})\cdot \mathrm {d} {\vec {r}}}$   or in index notation   ${\displaystyle \mathrm {d} F_{i}=\sum _{j}{\frac {\partial F_{i}}{\partial x_{j}}}\mathrm {d} x_{j}}$

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. ${\displaystyle \operatorname {grad} {\vec {A}}}$

The following applies to all constants and vector fields : ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle {\vec {A}},\,{\vec {B}}\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$

Linearity

${\displaystyle \operatorname {grad} (c\cdot {\vec {A}})=c\cdot \operatorname {grad} {\vec {A}}}$

${\displaystyle \operatorname {grad} ({\vec {A}}+{\vec {B}})=\operatorname {grad} {\vec {A}}+\operatorname {grad} {\vec {B}}}$

Product rule

${\displaystyle ({\vec {A}}\cdot \nabla ){\vec {B}}=(\operatorname {grad} {\vec {B}})\cdot {\vec {A}}}$

${\displaystyle \operatorname {grad} ({\vec {A}}\cdot {\vec {B}})=(\operatorname {grad} {\vec {A}})^{T}\cdot {\vec {B}}+(\operatorname {grad} {\vec {B}})^{T}\cdot {\vec {A}}}$

${\displaystyle \operatorname {grad} ({\vec {A}}^{\,2})=2\,(\operatorname {grad} {\vec {A}})^{T}\cdot {\vec {A}}}$

The above relationship can be transformed especially for vector fields : ${\displaystyle {\vec {A}},\,{\vec {B}}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}$

${\displaystyle \operatorname {grad} ({\vec {A}}\cdot {\vec {B}})=({\vec {B}}\cdot \nabla ){\vec {A}}+{\vec {B}}\times (\nabla \times {\vec {A}})+({\vec {A}}\cdot \nabla ){\vec {B}}+{\vec {A}}\times (\nabla \times {\vec {B}})}$

${\displaystyle \operatorname {grad} ({\vec {A}}^{\,2})=2({\vec {A}}\cdot \nabla ){\vec {A}}+2{\vec {A}}\times (\nabla \times {\vec {A}})}$

Examples

${\displaystyle \operatorname {grad} {\vec {r}}=I,}$where is the identity matrix . ${\displaystyle I}$

${\displaystyle \left(\operatorname {grad} {\frac {\vec {r}}{r^{3}}}\right)^{T}=\nabla \otimes {\frac {\vec {r}}{r^{3}}}=\left(\nabla {\frac {1}{r^{3}}}\right)\otimes {\vec {r}}+{\frac {1}{r^{3}}}(\nabla \otimes {\vec {r}})=-{\frac {3}{r^{5}}}{\vec {r}}\otimes {\vec {r}}+{\frac {1}{r^{3}}}I=-{\frac {1}{r^{5}}}(3{\vec {r}}\otimes {\vec {r}}-r^{2}I)}$

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".