scalar function

(noun)

any function whose domain is a vector space and whose value is its scalar field

Related Terms

Examples of scalar function in the following topics:

  • Line Integrals

    • A line integral is an integral where the function to be integrated is evaluated along a curve.
    • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
    • 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).
    • 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:

  • Linear Vector Spaces

    • Definition of Linear Vector Space: A linear vector space over a field $F$ of scalars is a set of elements $V$ together with a function called addition from $V \times V$ into $V$ (the Cartesian product $A \times B$ of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$ ) and a function called scalar multiplication from $F \times V$ into $V$ satisfying the following conditions for all $x,y,z \in V$ and all $\alpha, \beta \in F$ :
    • Addition and scalar multiplication are defined component-wise:
    • Functions themselves are vectors according to this definition.
    • Consider the space of functions mapping some nonempty set onto the scalars, with addition and multiplication defined by:
    • We use the square brackets to separate the function from its arguments.

  • 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 $\varphi$: $\nabla \times \nabla \varphi=0$.

  • Functions of Several Variables

    • As we will see, multivariable functions may yield counter-intuitive results when applied to limits and continuity.
    • Unlike a single variable function $f(x)$, for which the limits and continuity of the function need to be checked as $x$ varies on a line ($x$-axis), multivariable functions have infinite number of paths approaching a single point.Likewise, the path taken to evaluate a derivative or integral should always be specified when multivariable functions are involved.
    • We have also studied theorems linking derivatives and integrals of single variable functions.
    • A scalar field shown as a function of $(x,y)$.
    • Extensions of concepts used for single variable functions may require caution.

  • Matrix and Vector Norms

    • For scalars, the obvious answer is the absolute value.
    • The absolute value of a scalar has the property that it is never negative and it is zero if and only if the scalar itself is zero.
    • A norm is a function from the space of vectors onto the scalars, denoted by $\| \cdot \|$ satisfying the following properties for any two vectors $v$ and $u$ and any scalar $\alpha$ :

  • Calculus of Vector-Valued Functions

    • A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.
    • A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
    • The input of a vector-valued function could be a scalar or a vector.
    • When taking the derivative of a vector function, the function should be treated as a group of individual functions.
    • 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).

  • Multiplying Vectors by a Scalar

    • While adding a scalar to a vector is impossible because of their different dimensions in space, it is possible to multiply a vector by a scalar.
    • A scalar, however, cannot be multiplied by a vector.
    • To multiply a vector by a scalar, simply multiply the similar components, that is, the vector's magnitude by the scalar's magnitude.
    • Multiplying vectors by scalars is very useful in physics.
    • For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.

  • Unit Vectors and Multiplication by a Scalar

    • Multiplying a vector by a scalar is the same as multiplying its magnitude by a number.
    • In addition to adding vectors, vectors can also be multiplied by constants known as scalars.
    • Examples of scalars include an object's mass, height, or volume.
    • When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar .
    • (iii) Increasing the mass (scalar) increases the force (vector).

  • Scalars vs. Vectors

    • Physical quantities can usually be placed into two categories, vectors and scalars.
    • In contrast, scalars require only the magnitude.
    • Scalars differ from vectors in that they do not have a direction.
    • Scalars are used primarily to represent physical quantities for which a direction does not make sense.
    • This video introduces the difference between scalars and vectors.

  • Introduction to Scalars and Vectors

    • Given this information, is speed a scalar or a vector quantity?
    • Speed is a scalar quantity.
    • Distance is an example of a scalar quantity.
    • Scalars are never represented by arrows.
    • (A comparison of scalars vs. vectors is shown in . )