The PDE interfaces use a slightly different notation, compared to other physics interfaces and the associated documentation. The difference lies in the definition of the symbol ∇, pronounced nabla or del. For the physics interface equation sections and nodes (see Advanced Physics, Study, and Results Sections), the ∇ symbol applied to a scalar or vector variable denotes the following coordinate system-independent gradient, divergence, and curl operations:

In the PDE interfaces, nabla/del is interpreted as the vector of partial derivatives:

The spatial coordinates are denoted x1, …, xn, where n represents the number of space dimensions. When applied to a scalar or vector in a Cartesian coordinate system, this definition leads to an expression that is identical in form to the gradient, divergence, or curl in the same coordinate system. The same does not apply, however, in curvilinear systems such as the one implied in an axisymmetric geometry.

For example, the divergence of a vector u = [ur, uz] in an axisymmetric cylindrical system is

while the PDE interface interpretation of nabla/del is:

In practice, this means that to correctly implement equations containing the gradient, divergence, or curl in an axisymmetric geometry, you must compensate for the missing factors related to the curvature of the coordinate system. In particular, note that you must typically multiply the entire equation, as well as its boundary conditions, by a volume factor — in an axisymmetric geometry, for example, with a factor r — in order to recast it into one of the equation forms in COMSOL Multiphysics.

where β1 ,  … ,  βn are the components of the vector β.