Scalar field

A scalar field in which the intensity is represented by different colors (see legend).

In the multivariate analysis , the vector analysis and the differential geometry is a scalar field (short scalar ) is a function that each point of a room, a real number ( scalar ) assigns z. B. a temperature .

Scalar fields are of great importance in the field description of physics and in multi-dimensional vector analysis .

definition

A scalar field maps every point of a manifold onto a scalar . ${\ displaystyle \ varphi}$ ${\ displaystyle p}$ ${\ displaystyle M}$ ${\ displaystyle \ varphi (p)}$

A distinction is made between real-valued scalar fields

{\ displaystyle \ varphi \ colon M \ to \ mathbb {R}}

and complex valued scalar fields

{\ displaystyle \ varphi \ colon M \ to \ mathbb {C}}

.

One speaks of a stationary scalar field if the function values only depend on the location. If they also depend on the time, it is a non-stationary scalar field .

Examples

Examples of scalar fields in physics are air pressure , temperature , density or general potentials (also known as scalar potentials ).

Operations

Important operations related to scalar fields are:

Gradient of a scalar field that is a vector field .
Directional derivation of a scalar field.

classification

In contrast to the scalar field, a vector field assigns a vector to each point. A scalar field is the simplest tensor field .

Individual evidence

↑ Ziya Şanal: Mathematics for Engineers: Fundamentals - Applications in Maple . Springer, 2015, ISBN 978-3-658-10642-3 , pp. 550 .
^ ^A ^b Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf: Theoretical Physics . Springer, 2014, ISBN 978-3-642-54618-1 , pp. 31, 35, 274 .
^ Paul C. Matthews: Vector Calculus (= Springer Undergraduate Mathematics Series ). Springer, 2000, ISBN 978-3-540-76180-8 , 1.6 Scalar fields and vector fields.
↑ ^a ^b ^c Hans Karl Iben: Tensor calculation - mathematics for engineers and natural scientists, economists and farmers . Springer, 2013, ISBN 978-3-322-84792-8 , 4.2 Gradient, divergence and rotation of tensor fields.
↑ Josef Betten: Elementary tensor calculation for engineers: With numerous exercises and fully developed solutions . Springer, 2013, ISBN 978-3-663-14139-6 , pp. 112 .

[sanal2015-1] Ziya Şanal: Mathematics for Engineers: Fundamentals - Applications in Maple . Springer, 2015, ISBN 978-3-658-10642-3 , pp. 550 .

[bartelmann2014-2] A ^b Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf: Theoretical Physics . Springer, 2014, ISBN 978-3-642-54618-1 , pp. 31, 35, 274 .

[3] Paul C. Matthews: Vector Calculus (= Springer Undergraduate Mathematics Series ). Springer, 2000, ISBN 978-3-540-76180-8 , 1.6 Scalar fields and vector fields.

[iben2013-4] Hans Karl Iben: Tensor calculation - mathematics for engineers and natural scientists, economists and farmers . Springer, 2013, ISBN 978-3-322-84792-8 , 4.2 Gradient, divergence and rotation of tensor fields.

[5] Josef Betten: Elementary tensor calculation for engineers: With numerous exercises and fully developed solutions . Springer, 2013, ISBN 978-3-663-14139-6 , pp. 112 .