Vector identities – Wikipedia

Posted on May 27, 2019 by lordneo

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In this article, we note

{displaystyle times }

$times$ For the vector product and · for the scalar product.

The following identities can be useful in vector analysis.

In this section, a , b , c And d represent any vectors of

{displaystyle mathbb {R} ^{n}}

${mathbb R}^{n}$ .

Table of Contents

Writing conventions [ modifier | Modifier and code ]

In this article, the following agreements are used; Note that the position (lifting or lowered) of the clues does not, here, matter here since we work in a Euclidean context. This nevertheless makes it possible to find the couplings more directly (a higher index associating with a lower index).

Scalar product [ modifier | Modifier and code ]

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The scalar product of two vectors a And b is noted

{displaystyle mathbf {a} cdot mathbf {b} }

In Einstein summons convention, this is written:

{Displaystyle Mathbf {a} cdot mathbf {b} = a_ {i} b^{i} = a^{i} b_ {i}}}

Vector product [ modifier | Modifier and code ]

The vector product of two vectors a And b is noted

{displaystyle mathbf {a} times mathbf {b} }

In Einstein summons convention, this is written:

{displaystyle (mathbf {a} times mathbf {b} )_{i}={epsilon _{i}}^{jk}a_{j}b_{k}}

Levi-Civite Symbole [ modifier | Modifier and code ]

An identity often returning to demonstrations using the Einstein summons agreement is as follows:

{displaystyle {epsilon _{ij}}^{k}{epsilon _{k}}^{lm}=delta _{i}^{l}delta _{j}^{m}-delta _{i}^{m}delta _{j}^{l}}

With

{displaystyle delta }

$delta$ Le Symbole the crooked cock.

Triple products [ modifier | Modifier and code ]

We have the following result on the mixed product:

${Displaystyle Mathbf {A} CDOT (Mathbf {B} Times Mathbf {C} )=Mathbf {B} CDOT (Mathbf {C {C {C {C {C {C) } )}$

Demonstration

In Einstein summons convention we:

{displaystyle mathbf {a} cdot (mathbf {b} times mathbf {c} )=a_{i}{epsilon ^{i}}_{jk}b^{j}c^{k}}

By permuting twice the indices of the symbol of Levi-Civita and by rearranging the terms we obtain in turn the following equivalent expressions:

First of all :

{displaystyle b^{j}{epsilon _{jk}}^{i}c^{k}a_{i}=mathbf {b} cdot (mathbf {c} times mathbf {a} )}

Second:

{Displaystyle C^{K}{{EPSILON _{KILON _{K}}}_{J J}A_{J JAL}A_{JJ}=MATHBF {C} CDOT (Mathbf {A} Times Mathbf {B} }

Identity is thus demonstrated.

{displaystyle mathbf {a} times (mathbf {b} times mathbf {c} )=(mathbf {c} times mathbf {b} )times mathbf {a} =mathbf {b} (mathbf {a} cdot mathbf {c} )-mathbf {c} (mathbf {a} cdot mathbf {b} )}

${mathbf a}times ({mathbf b}times {mathbf c})=({mathbf c}times {mathbf b})times {mathbf a}={mathbf b}({mathbf a}cdot {mathbf c})-{mathbf c}({mathbf a}cdot {mathbf b})$

The first equality stems from the properties of the vector product:

{displaystyle (mathbf {a} times mathbf {b} )=-(mathbf {b} times mathbf {a} )}

$({mathbf a}times {mathbf b})=-({mathbf b}times {mathbf a})$ . The second is demonstrated below.

Demonstration

In Einstein summons convention we:

{DisplaySyle (MathBF {a} times (MathBF {b} times MathBF {c})) _ {i} = {}} ^ {} {} {}}} {_ _}} ^ ^ ^ _ _ št} } B_ {l} c_ {n}}

Using the properties of the Levi-Civita symbol and the Kronecker symbol, the right member can be rewritten as follows:

{displaystyle delta _{i}^{l}delta ^{jm}a_{j}b_{l}c_{m}-delta _{i}^{m}delta ^{jl}a_{j}b_{l}c_{m}=b_{i}a_{j}c^{j}-c_{i}a_{j}b^{j}}

By explaining the right member we find the identity:

{displaystyle (mathbf {a} times (mathbf {b} times mathbf {c} ))_{i}=b_{i}(mathbf {a} cdot mathbf {c} )-c_{i}(mathbf {a} cdot mathbf {b} )}

Others products [ modifier | Modifier and code ]

The identity of Binet-Cauchy:

{Displaystyle (Mathbf {A} Times Mathbf {B} )CDOT (Mathbf {C} TIMES MATHBF {D})=(Mathbf {A} CDOT MATHBF {C} )(Mathbf {B} CDOT Mathbf {D}-( Mathbf {A} CDOT Mathbf {D} )(Mathbf {B} CDOT Mathbf {C}}}

Note that we find the identity of Lagrange if a = c you if b = d .

Demonstration

In Einstein summons convention we:

{Displaystyle (Mathbf {A} Times Mathbf {B} )CDOT (MATHBF {C} TIMES MATHBF {D})={EPSILON _{ILOON _{I JKE}A_{JK}B_{K}{KLON ^{III }}_{lm}c^{l}d^{m}}

Using the properties of the Levi-Civita symbol and the Kronecker symbol, the right member can be rewritten as follows:

{Displaystyle delta _ {l}^{J} Delta _ {m}^{k} a_ {J} b_ {k} c^{l} d^{m} -DELTA _ {m}^{J} Delta _ {l}^{k} a_ {j} b_ {k} c^{l} d^{m} = a_ {l} c^{l} b_ {k} d^{k} -A_ ^{j} b_ {k} c^{k}}

The second member being obtained by simplifying and rearranging the terms. We find in this right -wing member the expression of scalar products and we finally have:

{Displaystyle (Mathbf {A} Times Mathbf {B} )CDOT (Mathbf {C} TIMES MATHBF {D})=(Mathbf {A} CDOT MATHBF {C} )(Mathbf {B} CDOT Mathbf {D}-( Mathbf {A} CDOT Mathbf {D} )(Mathbf {B} CDOT Mathbf {C}}}

This section provides an explicit list of the meaning of the symbols used for more clarity.

Divergence [ modifier | Modifier and code ]

Divergence of a vector field [ modifier | Modifier and code ]

For a vector field

{displaystyle mathbf {V} }

${mathbf V}$ , we generally write divergence as follows:

{displaystyle operatorname {mathbf {div}} (mathbf {v}) = nable cdot mathbf {v}}

It is a scalar field.

In Einstein summons convention, the divergence of a vector field is written:

{displaystyle nabla cdot mathbf {V} =partial _{i}V^{i}}

Divergence of a tensor [ modifier | Modifier and code ]

For a tensor

{displaystyle {stackrel {mathbf {mathfrak {T}} }{}}}

${stackrel {{mathbf {{mathfrak {T}}}}}{}}$ , we generally write divergence as follows:

{displaystyle operatorname {mathbf {div} } (mathbf {mathfrak {T}} )=nabla cdot mathbf {mathfrak {T}} }

Like the reduced divergence by 1 the order of the tensor, if

{displaystyle {stackrel {mathbf {mathfrak {T}} }{}}}

${stackrel {{mathbf {{mathfrak {T}}}}}{}}$ is of order 2 We would have a vector which is a tensor of order 1.

Rotationnel [ modifier | Modifier and code ]

For a vector field

{displaystyle mathbf {V} }

${mathbf V}$ , we generally write the rotational as follows:

{displaystyle operatorname {mathbf {rot}} (mathbf {v}) = Nable times mathbf {v}}

It is a vector field.

In Einstein summons convention, the rotational vector field is written:

{displaystyle (nabla times mathbf {V} )_{i}={epsilon _{i}}^{jk}partial _{j}V_{k}}

Gradient [ modifier | Modifier and code ]

Gradient of a vector field [ modifier | Modifier and code ]

For a vector field

{displaystyle mathbf {V} }

${mathbf V}$ , we usually write the gradient as follows:

{displaystyle operatorname {mathbf {grad}} (mathbf {v}) = nable mathbf {v}}

He is a tensor.

Gradient of a scalar field [ modifier | Modifier and code ]

For a scalar field

{displaystyle psi }

$psi$ , we usually write the gradient as follows:

{Displaystyle Operatorname {mathbf {grad}} (psi) = nabla psi}

It is a vector.

In Einstein’s summons convention, the gradient of a scalar field is written:

{Displaystyle (nabla psi) _ {i} = partial _ {i} dog}

Gradient rotational [ modifier | Modifier and code ]

The rotational of the gradient of any scalar

{displaystyle psi }

$psi$ is always zero:

{displaystyle nabla times (nable psi) = mathbf {0}}

Demonstration

In Einstein summons convention we:

{displaystyle left(nabla times (nabla psi )right)_{i}={epsilon _{i}}^{jk}partial _{j}partial _{k}psi }

By swapming the J and K clues (unhapping permutation) we obtain the equivalent expression:

{displaystyle {-epsilon _{i}}^{kj}partial _{k}partial _{j}psi =-left(nabla times (nabla psi )right)_{i}}

The change of sign comes from the unclean permutation of the clues. So we finally have:

{displaystyle left(nabla times (nabla psi )right)_{i}=-left(nabla times (nabla psi )right)_{i}Leftrightarrow left(nabla times (nabla psi )right)_{i}=0}

Identity is thus demonstrated.

Divergence of the Rotational [ modifier | Modifier and code ]

The divergence of the rotational of any vector field

{displaystyle mathbf {V} }

${mathbf V}$ is always zero:

{displaystyle nabla cdot (nabla times mathbf {V} )=0}

Demonstration

In Einstein summons convention we:

{displaystyle left(nabla cdot (nabla times mathbf {V} )right)=partial _{i}{epsilon ^{i}}_{jk}partial ^{j}V^{k}}

By swapming the indices I and J (unclean permutation) we obtain the equivalent expression:

{displaystyle -partial ^{j}{{epsilon _{j}}^{i}}_{k}partial _{i}V^{k}=-left(nabla cdot (nabla times mathbf {V} )right)}

The change of sign comes from the unclean permutation of the clues. So we finally have:

{displaystyle left(nabla cdot (nabla times mathbf {V} )right)=-left(nabla cdot (nabla times mathbf {V} )right)Leftrightarrow left(nabla cdot (nabla times mathbf {V} )right)=0}

Identity is thus demonstrated.

Laplacien [ modifier | Modifier and code ]

Laplacian of a scalar field [ modifier | Modifier and code ]

The Laplacian of a scalar field

{displaystyle psi }

$psi$ is defined as the divergence of the gradient:

{Displaystyle Nabla CDOT (Nabla Psi) = Nabla ^{2} psi}

It is a scalar field.

In Einstein summons, the Laplacian of a scalar field is noted as follows:

}

Laplacian of a vector field [ modifier | Modifier and code ]

The vector laplacian of a vector field is the vector whose components are the Laplacian of the components.

In Einstein summons convention, this is noted:

{DisplayStyle (Nabla {2} Mathbf {v}) _ {i} = ubla {2} (v_ {i}) = partial _ {j} partial {j} (v_ {i})}

Rotational Rotational [ modifier | Modifier and code ]

The rotational of the rotational vector field

{displaystyle mathbf {V} }

${mathbf V}$ is given by:

{displaystyle nabla times (Nabla Times mathbf {v}) = nable (nable cdot mathbf {v}) -nabla ^{2} mathbf {v}}

Demonstration

In Einstein summons convention we:

{displaystyle left(nabla times (nabla times mathbf {V} )right)_{i}={epsilon _{i}}^{jk}partial _{j}{epsilon _{k}}^{lm}partial _{l}V_{m}}

Using the properties of the Levi-Civita symbol and those of the Kronecker symbol, we then get:

{displaystyle delta _{i}^{l}delta ^{jm}partial _{j}partial _{l}V_{m}-delta _{i}^{m}delta ^{jl}partial _{j}partial _{l}V_{m}=partial _{i}partial ^{m}V_{m}-partial _{j}partial ^{j}V_{i}}

We find in the right member of this last expression the gradient of divergence and the Laplacian. So we finally have:

{displaystyle left(nabla times (nabla times mathbf {V} )right)_{i}=left(nabla (nabla cdot mathbf {V} )right)_{i}-nabla ^{2}(V_{i})}

Identity is thus demonstrated.

Field vector product by its rotational [ modifier | Modifier and code ]

The vector product of the field

{displaystyle mathbf {V} }

${mathbf V}$ by its rotational is given by:

{displaystyle mathbf {v} times (Nabla Times mathbf {v}) = nable left ({frac {mathbf {v} ^{2}} {2}} right)-(mathbf {v} cdot mathbf {nabla}) mathbf {V}}

Demonstration

In Einstein summons convention we:

{displaystyle left(mathbf {V} times (nabla times mathbf {V} )right)_{i}={epsilon _{i}}^{jk}V_{j}{epsilon _{k}}^{lm}partial _{l}V_{m}}

Using the properties of the Levi-Civita symbol and those of the Kronecker symbol, we then get:

{Displaystyle (Delta _ {i} {l} Delta _ {j} {m} -delta _ _ {i} {m} Delta _ {j {l}) v_ {j} V_ {m} = v_ {j} partial _ {i} v {j} -v_ {j} partial {j} v_}}}

Identity is thus demonstrated.

Other identities involving operators [ modifier | Modifier and code ]

In this section,

{displaystyle psi }

$psi$ And

{displaystyle phi }

$phi$ represent scalar fields,

{displaystyle mathbf {V} ,mathbf {A} }

${mathbf V},{mathbf A}$ And

{displaystyle mathbf {B} }

${mathbf B}$ represent vector fields.

${Displaystyle Nabla (Psi Phi) = (Nabla Psi) Phi +(Nabla Phi) PSI}$

This relationship immediately stems from the product rule.

${displaystyle nabla cdot (psi mathbf {v}) = (nable psi) cdot mathbf {v} +(nable cdot mathbf {v}) psi}$

Demonstration

In Einstein summons convention, we have:

{displaystyle nabla cdot (psi mathbf {v}) = partial _ {i} (psi mathbf {v})^{I}}

The product rule being application, this last term is equivalent to:

{Displaystyle (partial _ {i} psi) v^{i} +psi (partial _ {i} v^{i}) = (nabla psi) cdot mathbf {v} +(nabla cdot mathbf {v}) psi}

Identity is thus demonstrated.

${displaystyle nabla times (psi mathbf {v}) = (nable psi) Times mathbf {v} +(Nabla times mathbf {v}) psi}$

Demonstration

In Einstein summons convention, we have:

{displaystyle (Nabla Times (psi mathbf {v})) _ {i} = {epsilon _ {i}}^{jk} partial _ {j} (psi mathbf {v}) _ {k}}

The product rule being application, this last term is equivalent to:

{Displaystylele {epsilon _ {i}}} {jk} (partial _} psi) v_ {k}+{epsilon _ {i}}} {jk} (partial _} v_ {k}) psi = (Nabla psi) Times mathbf {v}) _ {i}+(Nabla Times mathbf {v}) _ {i} psi}

Identity is thus demonstrated.

Gradient of a scalar product [ modifier | Modifier and code ]

${displaystyle nabla (mathbf {a} cdot mathbf {b}) = (mathbf {a} cdot nabla) mathbf {b} +(mathbf {b} cdot nabla) mathbf {a} +mathbf {a} times (Nabla times mathbf {B})+mathbf {b} times (Nabla Times mathbf {a})}$

Divergence from a vector product [ modifier | Modifier and code ]

${Displaystyle Nabla CDOT (Mathbf {A} TIMES MATHBF {B} )=(Nabla Times Mathbf {A} )Cdot Mathbf {B} -Mathbf {A} CDOt (Nabla Times Mathbf {B}}$

Demonstration

In Einstein summons convention, we have:

{displaystyle (nabla cdot (mathbf {A} times mathbf {B} ))=partial _{i}({epsilon ^{i}}_{jk}A^{j}B^{k})}

The product rule being applied, this last term is equal to:

{dysplaystylele {epsilon {i}} _ {jk} (partial _ {i} a {j}) b^{k}+{epsilon^{i}} _ {jk} a^{j} (partial _ {i} b^{k})}

By performing a permutation pair of index on the first term and an odd on the second, we get:

{displaystyle ({{Epsilon _{k}}^{i}_{j}partial _{I^j})b^{k}-a^{j}({Epsilon _{j} }^{I}}_{k}partial _{I}b^{k})=(Nlabla Times Mathbf {} )cdot mathbf {b} -mathbf {ag} cdot (Natla Times Mathbf {b} )}}.

The change of sign comes from the unclean permutation of the symbol of Levi-Civita.

Identity is thus demonstrated.

Rotational of a vector product [ modifier | Modifier and code ]

${displaystyle nabla times (mathbf {a} times mathbf {b}) = (nable cdot mathbf {b}) mathbf {a} -(nable cdot mathbf {a}) mathbf {b} + +(mathbf {b} cdot nabla) mathbf {a} -(mathbf {a} cdot nabla) mathbf {b}}$

Demonstration

In Einstein summons convention, we have:

{displaystyle (nabla times (mathbf {A} times mathbf {B} ))_{i}={epsilon _{i}}^{jk}partial _{j}({epsilon _{k}}^{lm}A_{l}B_{m})=epsilon _{i}^{jk}epsilon _{k}^{lm}(partial _{j}A_{l}B_{m}+A_{l}partial _{j}B_{m})}

where the product rule was used. With the properties of the Levi-Civita symbol, the latter term rewritten

{displaystyle (delta ^{il}delta ^{jm}-delta ^{im}delta ^{jl})(partial _{j}A_{l}B_{m}+A_{l}partial _{j}B_{m})=partial _{j}A_{i}B^{j}+A_{i}partial _{j}B^{j}-partial _{j}A^{j}B_{i}-A^{j}partial _{j}B_{i}}

The right -wing member can then be written as follows:

{displaystyle (mathbf {b} cdot nabla) a_ {i}+a_ {i} (nable cdot mathbf {b})-(nable cdot mathbf {a}) b_ {i}-(mathbf {a} cdot nabla) b_ b_ {I}}

Identity is thus demonstrated.

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Vector identities – Wikipedia

Writing conventions [ modifier | Modifier and code ]

Scalar product [ modifier | Modifier and code ]

Vector product [ modifier | Modifier and code ]

Levi-Civite Symbole [ modifier | Modifier and code ]

Triple products [ modifier | Modifier and code ]

Others products [ modifier | Modifier and code ]

Divergence [ modifier | Modifier and code ]

Divergence of a vector field [ modifier | Modifier and code ]

Divergence of a tensor [ modifier | Modifier and code ]

Rotationnel [ modifier | Modifier and code ]

Gradient [ modifier | Modifier and code ]

Gradient of a vector field [ modifier | Modifier and code ]

Gradient of a scalar field [ modifier | Modifier and code ]

Gradient rotational [ modifier | Modifier and code ]

Divergence of the Rotational [ modifier | Modifier and code ]

Laplacien [ modifier | Modifier and code ]

Laplacian of a scalar field [ modifier | Modifier and code ]

Laplacian of a vector field [ modifier | Modifier and code ]

Rotational Rotational [ modifier | Modifier and code ]

Field vector product by its rotational [ modifier | Modifier and code ]

Other identities involving operators [ modifier | Modifier and code ]

Gradient of a scalar product [ modifier | Modifier and code ]

Divergence from a vector product [ modifier | Modifier and code ]

Rotational of a vector product [ modifier | Modifier and code ]

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