We shall introduce now a scalar function by the relationship.

A scalar function called the electric potential can help.

It is however an algebra over the ring of scalar functions.

This variable X is a scalar function of position in spacetime.

One-dimensional points just define a scalar function of the parameter.

This is a Poisson equation for the scalar function .

In general, the scalar function can help.

Scalar functions take a single value, perform an operation and return a single value.

This definition is analogous to a concave scalar function.

It can be used to calculate directional derivatives of scalar functions or normal directions.

The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions.

The following relationship exists for scalar functions :

Let denote the nonlinearity, a scalar function with non-negative output.

A potential function is a scalar function that defines how the vectors will behave.

As a result, can be represented as the gradient of a scalar function :

Under conformal rescaling of the metric for some scalar function .

Each vector generates a scalar function by the formula:

It describes the speeds of the charge carriers that have a density described by the scalar function ρ.

In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields.

The second condition is a requirement of F so that it can be expressed as the gradient of a scalar function.

The components of a conservative force are given by where φ is the potential, which is a scalar function of position only.

If we add to A the gradient of a scalar function, the resulting vector A' still gives the same magnetic field because.

A scalar function whose contour lines define the streamlines is known as the stream function.

We found there that by choosing a scalar function in the form we could automatically satisfy the other equation for the electric field strength,.

A scalar function is computationally and conceptually easier to deal with than a vector function.

The general expressions for a scalar-valued function, f, are a little simpler.

This is analogous to the notion of strictly increasing for scalar-valued functions of one scalar argument.

Value entities may have attributes, but they are restricted to single- and scalar-valued functions, and they do not have a key.

Dual to scalar-valued functions - maps - are maps which correspond to curves or paths in a manifold.

Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system.

A function is Pettis integrable (over ) if the scalar-valued function is integrable for every functional .

Mathematically, the quantity is not a vector because it is a positive scalar-valued function of the prescribed direction and sense, in this example, of the downward vertical.

Once a reference frame has been chosen, the derivative of a vector-valued function can be computed using techniques similar to those for computing derivatives of scalar-valued functions.

This theorem has a powerful converse; namely, if F is a path-independent vector field, then F is the gradient of some scalar-valued function.

In other words, the Jacobian for a scalar-valued multivariable function is the gradient and that of a scalar-valued function of single variable is simply its derivative.

The scalar approach defines flux density as a scalar-valued function of a direction and sense in space prescribed by the investigator at a point prescribed by the investigator.

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

The Laplace-Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the Laplace transform of a Stieltjes measure.

A basic example of maps between manifolds are scalar-valued functions on a manifold, or sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra.

The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable.

Thus the converse can alternatively be stated as follows: If the integral of F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.

Whereas for scalar-valued functions there is only a single possible reference frame, to take the derivative of a vector-valued function requires the choice of a reference frame (at least when a fixed Cartesian coordinate system is not implied as such).

The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see incidence algebra for the details.

For Bézier curves, it has become customary to refer to the -vectors p in a parametric representation p of a curve or surface in -space as control points, while the scalar-valued functions , defined over the relevant parameter domain, are the corresponding weight or blending functions.

The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e, if F is conservative), then F is a path-independent vector field (i.e, the integral of F over some piecewise-differentiable curve is dependent only on end points).

In an inhomogeneous non-isotropic radiative field, the spectral flux density defined as a scalar-valued function of direction and sense contains much more directional information than does the spectral flux density defined as a vector, but the full radiometric information is customarily stated as the spectral radiance (or specific intensity).