scalar

(noun)

a quantity that has magnitude but not direction; compare vector

Related Terms

Examples of scalar in the following topics:

  • Conservative Vector Fields

    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • Definition: A vector field $\mathbf{v}$ is said to be conservative if there exists a scalar field $\varphi$ such that $\mathbf{v}=\nabla\varphi$.
    • When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf{v}$.
    • For any scalar field $\varphi$: $\nabla \times \nabla \varphi=0$.

  • Line Integrals

    • The function to be integrated may be a scalar field or a vector field.
    • The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
    • More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve.
    • For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:
    • The line integral over a scalar field $f$ can be thought of as the area under the curve $C$ along a surface $z = f(x,y)$, described by the field.

  • Surface Integrals of Vector Fields

    • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
    • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
    • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
    • Based on this reasoning, to find the flux, we need to take the dot product of $\mathbf{v}$ with the unit surface normal to $S$, at each point, which will give us a scalar field, and integrate the obtained field as above.
    • Explain relationship between surface integral of vector fields and surface integral of a scalar field

  • Fundamental Theorem for Line Integrals

    • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
    • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
    • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

  • Physics and Engineering: Fluid Pressure and Force

    • Pressure is a scalar quantity.
    • The pressure is the scalar proportionality constant that relates the two normal vectors:

  • Functions of Several Variables

    • A scalar field shown as a function of $(x,y)$.

  • Parametric Surfaces and Surface Integrals

    • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

  • Stokes' Theorem

    • Therefore, electric field can be written as a gradient of a scalar field:
    • The scalar field $\varphi$ in the case of electromagnetism is called electric potential.

  • Calculus of Vector-Valued Functions

    • The input of a vector-valued function could be a scalar or a vector.
    • Vector functions are used in a number of differential operations, such as gradient (measures the rate and direction of change in a scalar field), curl (measures the tendency of the vector function to rotate about a point in a vector field), and divergence (measures the magnitude of a source at a given point in a vector field).

  • Vector Fields

    • Gradient field: Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).
    • A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that: