Vector area is unit normal times scalar area

Vector area, or oriented area, is a vector quantity equal to the surface integral of the surface normal. If the unit normal is constant at all points of the surface, integral can be reduced to a product of the unit normal to the scalar area of the surface.

Notes:

1. If the normal changes direction across the surface, you can divide the surface into parts of constant unit normal and sum up the vector areas of all parts:

   \[\vec A = \sum_i \vec A_i = \sum_i \vec n_i A_i\]

   Alternatively, use the surface integral that sums up infinitesimal vector areas across the surface:

   \[\vec A = \iint \limits_S d \vec A = \iint \limits_S \vec n (\vec r) dA\]

Conditions:

1. The surface is bounded (i.e. finite).

2. The surface normal is the same thoughout the given region, which can be achieved by choosing a small enough (or infinitesimal) surface.

Links:

1. Wikipedia — Vector area.

vector_area

Vector area pertaining to the given region.

Symbol:

A

Latex:

\({\vec A}\)

Dimension:

area

unit_normal

Unit vector normal to the surface.

Notes:

1. \(\left \Vert \vec n \right \Vert = 1\), i.e. it has unit magnitude.

Symbol:

n

Latex:

\({\vec n}\)

Dimension:

dimensionless

scalar_area

Scalar area of the given region.

Symbol:

A

Latex:

\(A\)

Dimension:

area

law

A = n * A

Latex:

\[{\vec A} = {\vec n} A\]