Scalar field
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In mathematics and physics, a scalar field associates a scalar to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure.
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Definition
A scalar field is a function from Rn to R. That is, it is a function defined on the n-dimensional Euclidean space with real values. Often it is required to be continuous, or one or more times differentiable, that is, a function of class Ck.
The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space.
The derivative of a scalar field results in a vector field called the gradient.
Examples found in physics
- Potential field
- In quantum field theory a scalar field is associated with spin 0 particles, like mesons. The scalar field may be real or complex valued (depending on whether it will associate a real or complex number to every point of space-time). Complex scalar fields represent charged particles.
Other kinds of fields
- Vector fields, which associate a vector to every point in space. Some examples of vector fields include the electromagnetic field and the Newtonian gravitational field.
- Tensor fields, which associate a tensor to every point in space. In general relativity, gravity is associated with a tensor field. In particular, with the Riemann curvature tensor. In Kaluza-Klein theory spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
Visualization of coordinates
Basically all you need to know about two different coordinate systems is how to get from one to the other, but it usually helps to visualize one as a rectangular grid. But you can also do that with polar coordinates. How weird Euclidean coordinates are when you do this. If there is more structure, such as a metric, then you know about the angles between your coordinate vectors and their lengths, and in that way you can choose which ones are more rectangular.
Differential geometry
A scalar field on a Ck-manifold is a Ck function to the real numbers. Taking Rn as manifold gives back the special case of vector calculus.
A scalar field is also a 0-form. See differential forms.de:Skalarfeld fr:Champ scalaire he:שדה סקלרי pl:Pole skalarne sv:Skalärfält zh:标量场
