Dual

The metric dual or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,

where $$\mathbf G$$ is the extended metric tensor. In projective geometric algebra, this dual is also called the bulk dual because it is the complement of the bulk components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .

The metric antidual or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

where $$\mathbb G$$ is the extended antimetric tensor. In projective geometric algebra, this dual is also called the weight dual because it is the complement of the weight components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .