Let X i be a basis of tangent vector fields not necessarily induced by a coordinate system | The operator can be extended to operate on tensors as the divergence of the covariant derivative |
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Proofs of all these statements may be found in the book by Isaac Chavel | Conversely, 2 characterizes the Laplace—Beltrami operator completely, in the sense that it is the only operator with this property |
In , the Laplace—Beltrami operator is a generalization of the to functions defined on in and, even more generally, on and.
28Not to be confused with | Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace—Beltrami operator itself does not depend on this additional structure |
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One can also give an intrinsic description of the Laplace—Beltrami operator on the sphere in a | In , such as or , one obtains |