note-math

Gradient and divergence are adjoint to each other.

or

Generalize to general tensors. For example, suppose "covariant differentiation" adds to the first position, i.e.

Then the adjoint is

let

adjoint :=

Which is the Lagrangian for Laplacian? We can arbitrarily add the action mass term .

• . eq or
• . eq or
• . eq or

Special case

adjoint

In coordinates

adjoint

are the basic building blocks for constructing mixed symmetry.

Using algebraic techniques, define (ref-24, p.34–35)

Using

Define

In coordinates

and

"product rule"

Then define the adjoint . Also there is

"product rule" (Question)

But , due to the symmetry of .

So is the exterior derivative.

In coordinates

also has an adjoint

The Lagrangian for the Laplacian of ?

• . eq
• . eq
• . eq

The variables in the Lagrangian should probably use instead of

The differentiation of the Lagrangian should probably use instead of

For scalar fields as zeroth-order tensor fields, assume and . At this time

In coordinates

• gradient

• divergence

• Laplacian

At the origin of geodesic coordinates,

• gradient where is the -th component of the quadratic form of .

• divergence

• harmonician

The same is true in flat metric coordinates.