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Pressure is a scalar quantity. It relates the vector area element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates these two normal vectors: = =.
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
In theoretical acoustics, [2] it is often desirable to work with the acoustic wave equation of the velocity potential ϕ instead of pressure p and/or particle velocity u. ∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = 0 {\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=0} Solving the wave equation for ...
This vector equation is one meaningful scalar equation and two 0 = 0 equations. The assumptions for the stream function equation are: The flow is incompressible and Newtonian.
Again the oscillatory part of the velocity vector v is related to the velocity potential by v = ∇φ, while as before Δ is the Laplace operator, and c is the average speed of sound in the homogeneous medium. Note that also the oscillatory parts of the pressure p and density ρ each individually satisfy the wave equation, in this approximation.
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity.Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields (known as scalar bosons), such as the Higgs field.