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vector Pressure gradient: Pressure per unit distance pascal/m L −2 M 1 T −2: vector Temperature gradient: steepest rate of temperature change at a particular location K/m L −1 Θ: vector Torque: τ: Product of a force and the perpendicular distance of the force from the point about which it is exerted
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.
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...
The advection equation is a first-order hyperbolic partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. [1] It is derived using the scalar field's conservation law, together with Gauss's theorem, and taking the infinitesimal limit.
For example, for a macroscopic scalar field φ(x, t) and a macroscopic vector field A(x, t) the definition becomes: +, +. In the scalar case ∇ φ is simply the gradient of a scalar, while ∇ A is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of A as a function of x ).
Given vector fields V, W defined on S and a smooth function f defined on S, the operations of scalar multiplication and vector addition, ():= () (+) ():= + (), make the smooth vector fields into a module over the ring of smooth functions, where multiplication of functions is defined pointwise.
In many engineering applications the local flow velocity vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity ¯ (with the usual dimension of length per time), defined as the quotient between the volume flow rate ˙ (with dimension of cubed length per time) and the cross sectional area (with dimension of square length):
By its own definition, the vorticity vector is a solenoidal field since = In a two-dimensional flow , ω {\displaystyle {\boldsymbol {\omega }}} is always perpendicular to the plane of the flow, and can therefore be considered a scalar field .