One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

where α_x is linear.

Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:

where the f_i are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.

Examples

Forma

Forma is a Latin word meaning "form". Both the word "forma" and the word "form" are used interchangeably as informal terms in biology:

Formwork

Formwork is the term given to either temporary or permanent molds into which concrete or similar materials are poured. In the context of concrete construction, the falsework supports the shuttering moulds.

Formwork and concrete form types

Formwork comes in several types:

Form (religion)

In academic discussions of organized religion, the term form is sometimes used to describe prescriptions or norms on religious practice.

Christian forms

Forms in Christianity are mostly familiarly dictates of church authority or tradition (e.g. church government, liturgy, doctrine). However, the term is used by some authors to refer to a broader category that includes other patterns of religious practice.

Most notably, Christian scholar D. G. Hart uses this term to compare and contrast the practices of evangelical Protestants and what he calls "confessional Protestants" (for example Anglicans and most Lutherans). He argues that the confessionals follow forms that are dictated by church authority or tradition, and calls these forms churchly forms. On the other hand, noting the resistance to such central authority and tradition among evangelicals, he labels the forms of these denominations parachurchly forms, as they are often dictated by parachurch organizations and other influences beyond the direct control of any particular church.