Vector triple product in 4 dimensions
$begingroup$
I have rewritten an equation in 3 dimensions
$$ vec{f} + nabla a + b nabla c = vec{0} $$
where $a,b,c$ are unknown functions to be determined and $vec{f}$ is a known vector by taking the curl
$$nabla times vec{f} + nabla b times nabla c = vec{0}$$
and using the the property of the triple product $vec{u} cdot (vec{u} times vec{v}) = 0$ to give 3 first-order linear partial differential equations, 1 for each of the unknown functions
$$ nabla b cdot (nabla times vec{f}) = 0 \ nabla c cdot (nabla times vec{f}) = 0 \ nabla a cdot (nabla times vec{f}) + vec{f} cdot (nabla times vec{f}) = 0 ; .$$
The 1st-order PDE's allowed me to reduce the number of independent variables to 2.
I am now faced with the same problem in 4 dimensions.
Question:
Is there a way of applying a similar method to derive 1st-order PDE's for each of the unknown functions?
Addendum:
I am thinking that this can be solved more easily using index notation. If the original equation is
$$ f_{k} + partial_{k} a + b partial_{k} c = 0_k $$
then one can perform the following steps (to obtain an equation for $b$)
$$ partial_j b , partial_l left( f_{k} + partial_{k} a + b partial_{k} c right) varepsilon^{i j k l} textbf{1}_i = 0 $$
where $textbf{1}_i$ is supposed to mean a vector of ones and $varepsilon^{i j k l}$ is the Levi-Civita symbol in 4 dimensions. Some terms drop out and and we are left with
$$ partial_j b , partial_l f_{k} varepsilon^{i j k l} textbf{1}_i = 0 ; .$$
Do you think this is correct?
pde partial-derivative vector-analysis
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
$endgroup$
add a comment |
$begingroup$
I have rewritten an equation in 3 dimensions
$$ vec{f} + nabla a + b nabla c = vec{0} $$
where $a,b,c$ are unknown functions to be determined and $vec{f}$ is a known vector by taking the curl
$$nabla times vec{f} + nabla b times nabla c = vec{0}$$
and using the the property of the triple product $vec{u} cdot (vec{u} times vec{v}) = 0$ to give 3 first-order linear partial differential equations, 1 for each of the unknown functions
$$ nabla b cdot (nabla times vec{f}) = 0 \ nabla c cdot (nabla times vec{f}) = 0 \ nabla a cdot (nabla times vec{f}) + vec{f} cdot (nabla times vec{f}) = 0 ; .$$
The 1st-order PDE's allowed me to reduce the number of independent variables to 2.
I am now faced with the same problem in 4 dimensions.
Question:
Is there a way of applying a similar method to derive 1st-order PDE's for each of the unknown functions?
Addendum:
I am thinking that this can be solved more easily using index notation. If the original equation is
$$ f_{k} + partial_{k} a + b partial_{k} c = 0_k $$
then one can perform the following steps (to obtain an equation for $b$)
$$ partial_j b , partial_l left( f_{k} + partial_{k} a + b partial_{k} c right) varepsilon^{i j k l} textbf{1}_i = 0 $$
where $textbf{1}_i$ is supposed to mean a vector of ones and $varepsilon^{i j k l}$ is the Levi-Civita symbol in 4 dimensions. Some terms drop out and and we are left with
$$ partial_j b , partial_l f_{k} varepsilon^{i j k l} textbf{1}_i = 0 ; .$$
Do you think this is correct?
pde partial-derivative vector-analysis
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
$endgroup$
add a comment |
$begingroup$
I have rewritten an equation in 3 dimensions
$$ vec{f} + nabla a + b nabla c = vec{0} $$
where $a,b,c$ are unknown functions to be determined and $vec{f}$ is a known vector by taking the curl
$$nabla times vec{f} + nabla b times nabla c = vec{0}$$
and using the the property of the triple product $vec{u} cdot (vec{u} times vec{v}) = 0$ to give 3 first-order linear partial differential equations, 1 for each of the unknown functions
$$ nabla b cdot (nabla times vec{f}) = 0 \ nabla c cdot (nabla times vec{f}) = 0 \ nabla a cdot (nabla times vec{f}) + vec{f} cdot (nabla times vec{f}) = 0 ; .$$
The 1st-order PDE's allowed me to reduce the number of independent variables to 2.
I am now faced with the same problem in 4 dimensions.
Question:
Is there a way of applying a similar method to derive 1st-order PDE's for each of the unknown functions?
Addendum:
I am thinking that this can be solved more easily using index notation. If the original equation is
$$ f_{k} + partial_{k} a + b partial_{k} c = 0_k $$
then one can perform the following steps (to obtain an equation for $b$)
$$ partial_j b , partial_l left( f_{k} + partial_{k} a + b partial_{k} c right) varepsilon^{i j k l} textbf{1}_i = 0 $$
where $textbf{1}_i$ is supposed to mean a vector of ones and $varepsilon^{i j k l}$ is the Levi-Civita symbol in 4 dimensions. Some terms drop out and and we are left with
$$ partial_j b , partial_l f_{k} varepsilon^{i j k l} textbf{1}_i = 0 ; .$$
Do you think this is correct?
pde partial-derivative vector-analysis
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
$endgroup$
I have rewritten an equation in 3 dimensions
$$ vec{f} + nabla a + b nabla c = vec{0} $$
where $a,b,c$ are unknown functions to be determined and $vec{f}$ is a known vector by taking the curl
$$nabla times vec{f} + nabla b times nabla c = vec{0}$$
and using the the property of the triple product $vec{u} cdot (vec{u} times vec{v}) = 0$ to give 3 first-order linear partial differential equations, 1 for each of the unknown functions
$$ nabla b cdot (nabla times vec{f}) = 0 \ nabla c cdot (nabla times vec{f}) = 0 \ nabla a cdot (nabla times vec{f}) + vec{f} cdot (nabla times vec{f}) = 0 ; .$$
The 1st-order PDE's allowed me to reduce the number of independent variables to 2.
I am now faced with the same problem in 4 dimensions.
Question:
Is there a way of applying a similar method to derive 1st-order PDE's for each of the unknown functions?
Addendum:
I am thinking that this can be solved more easily using index notation. If the original equation is
$$ f_{k} + partial_{k} a + b partial_{k} c = 0_k $$
then one can perform the following steps (to obtain an equation for $b$)
$$ partial_j b , partial_l left( f_{k} + partial_{k} a + b partial_{k} c right) varepsilon^{i j k l} textbf{1}_i = 0 $$
where $textbf{1}_i$ is supposed to mean a vector of ones and $varepsilon^{i j k l}$ is the Levi-Civita symbol in 4 dimensions. Some terms drop out and and we are left with
$$ partial_j b , partial_l f_{k} varepsilon^{i j k l} textbf{1}_i = 0 ; .$$
Do you think this is correct?
pde partial-derivative vector-analysis
pde partial-derivative vector-analysis
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
edited Dec 20 '18 at 22:37
Crenguta
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
asked Dec 19 '18 at 18:57
CrengutaCrenguta
456
456
add a comment |
add a comment |
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