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mathematical physics > differential operator > vector Laplace operator

mathematical analysis > functional analysis > operator > differential operator > vector Laplace operator

mathematical physics > vector calculus > vector calculus identities > vector Laplace operator

mathematical analysis > calculus > vector calculus > vector calculus identities > vector Laplace operator

geometry > differential geometry > vector calculus > vector calculus identities > vector Laplace operator

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vector Laplace operator

The vector Laplace operator, also denoted by ${\\displaystyle \ abla ^{2}}$ , is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.
The vector Laplacian of a vector field ${\\displaystyle \\mathbf {A} }$ is defined as
${\\displaystyle \ abla ^{2}\\mathbf {A} =\ abla (\ abla \\cdot \\mathbf {A} )-\ abla \ imes (\ abla \ imes \\mathbf {A} ).}$
In Cartesian coordinates, this reduces to the much simpler form as
${\\displaystyle \ abla ^{2}\\mathbf {A} =(\ abla ^{2}A_{x},\ abla ^{2}A_{y},\ abla ^{2}A_{z}),}$
where ${\\displaystyle A_{x}}$ , ${\\displaystyle A_{y}}$ , and ${\\displaystyle A_{z}}$ are the components of the vector field ${\\displaystyle \\mathbf {A} }$ , and ${\\displaystyle \ abla ^{2}}$ just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.
For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Laplace_operator#Vector_Laplacian)

laplacien vectoriel

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