In this article, we note
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
.
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 ]
The scalar product of two vectors a And b is noted
-
In Einstein summons convention, this is written:
-
Vector product [ modifier | Modifier and code ]
The vector product of two vectors a And b is noted
-
In Einstein summons convention, this is written:
-
Levi-Civite Symbole [ modifier | Modifier and code ]
An identity often returning to demonstrations using the Einstein summons agreement is as follows:
-
With
Le Symbole the crooked cock.
Triple products [ modifier | Modifier and code ]
We have the following result on the mixed product:
Demonstration
In Einstein summons convention we:
-
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 :
-
Second:
-
Identity is thus demonstrated.
The first equality stems from the properties of the vector product:
. The second is demonstrated below.
Demonstration
In Einstein summons convention we:
-
Using the properties of the Levi-Civita symbol and the Kronecker symbol, the right member can be rewritten as follows:
-
By explaining the right member we find the identity:
-
Others products [ modifier | Modifier and code ]
The identity of Binet-Cauchy:
-
Note that we find the identity of Lagrange if a = c you if b = d .
Demonstration
In Einstein summons convention we:
-
Using the properties of the Levi-Civita symbol and the Kronecker symbol, the right member can be rewritten as follows:
-
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:
-
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
, we generally write divergence as follows:
-
It is a scalar field.
In Einstein summons convention, the divergence of a vector field is written:
-
Divergence of a tensor [ modifier | Modifier and code ]
For a tensor
, we generally write divergence as follows:
-
Like the reduced divergence by 1 the order of the tensor, if
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
, we generally write the rotational as follows:
-
It is a vector field.
In Einstein summons convention, the rotational vector field is written:
-
Gradient [ modifier | Modifier and code ]
Gradient of a vector field [ modifier | Modifier and code ]
For a vector field
, we usually write the gradient as follows:
-
He is a tensor.
Gradient of a scalar field [ modifier | Modifier and code ]
For a scalar field
, we usually write the gradient as follows:
-
It is a vector.
In Einstein’s summons convention, the gradient of a scalar field is written:
-
Gradient rotational [ modifier | Modifier and code ]
The rotational of the gradient of any scalar
is always zero:
-
Demonstration
In Einstein summons convention we:
-
By swapming the J and K clues (unhapping permutation) we obtain the equivalent expression:
-
The change of sign comes from the unclean permutation of the clues. So we finally have:
-
Identity is thus demonstrated.
Divergence of the Rotational [ modifier | Modifier and code ]
The divergence of the rotational of any vector field
is always zero:
-
Demonstration
In Einstein summons convention we:
-
By swapming the indices I and J (unclean permutation) we obtain the equivalent expression:
-
The change of sign comes from the unclean permutation of the clues. So we finally have:
-
Identity is thus demonstrated.
Laplacien [ modifier | Modifier and code ]
Laplacian of a scalar field [ modifier | Modifier and code ]
The Laplacian of a scalar field
is defined as the divergence of the gradient:
-
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:
-
Rotational Rotational [ modifier | Modifier and code ]
The rotational of the rotational vector field
is given by:
-
Demonstration
In Einstein summons convention we:
-
Using the properties of the Levi-Civita symbol and those of the Kronecker symbol, we then get:
-
We find in the right member of this last expression the gradient of divergence and the Laplacian. So we finally have:
-
Identity is thus demonstrated.
Field vector product by its rotational [ modifier | Modifier and code ]
The vector product of the field
by its rotational is given by:
-
Demonstration
In Einstein summons convention we:
-
Using the properties of the Levi-Civita symbol and those of the Kronecker symbol, we then get:
-
Identity is thus demonstrated.
Other identities involving operators [ modifier | Modifier and code ]
In this section,
And
represent scalar fields,
And
represent vector fields.
This relationship immediately stems from the product rule.
Demonstration
In Einstein summons convention, we have:
-
The product rule being application, this last term is equivalent to:
-
Identity is thus demonstrated.
Demonstration
In Einstein summons convention, we have:
-
The product rule being application, this last term is equivalent to:
-
Identity is thus demonstrated.
Gradient of a scalar product [ modifier | Modifier and code ]
Divergence from a vector product [ modifier | Modifier and code ]
Demonstration
In Einstein summons convention, we have:
-
The product rule being applied, this last term is equal to:
-
By performing a permutation pair of index on the first term and an odd on the second, we get:
-
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 ]
Demonstration
In Einstein summons convention, we have:
-
where the product rule was used. With the properties of the Levi-Civita symbol, the latter term rewritten
-
The right -wing member can then be written as follows:
-
Identity is thus demonstrated.
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