Divergence Theorem

The divergence theorem states that the volume integral of the divergence of a vector field over a volume bounded by a surface is equal to the surface integral of the vector field projected on the outward facing normal of the surface .

Product Rule

Product rule for the product of a scalar and a vector is useful to reduce the derivative order on an expression in conjunction with the divergence theorem.

Shuffle the terms (and note that this is valid for a vector and a scalar as well)

The right most term () can be transformed using the divergence theorem. This can be used to effectively shift a derivative over to the test function when building a residual.

Fundamental Lemma of calculus of variations

For a functional

the functional derivative in the Cahn-Hilliard equation can be calculated using the rule

Note that the above formula is only valid up to first order derivatives (i.e. ). The general formula (required for some more advanced phase field models) for a functional

with higher order derivatives is

where the vector , and is a tensor whose components are partial derivative operators of order

Weak form of the ACInterface Kernel

The term is multiplied with the test function and integrated, yielding

we moved the over to the right and identify a vector term and a scalar term . Then we use the third equality in the _Product rule_ section to obtain

The last term is converted into a surface integral using the _Divergence theorem_, yielding