is the curvature scalar.

where R ( g ) is the scalar curvature and vol ( g ) is the volume element.

The scalar curvature, written in components, then expands to

In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness.

Two more generalizations of curvature are the scalar curvature and Ricci curvature.

This difference ( in a suitable limit ) is measured by the scalar curvature.

Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions.

It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

In Kaluza-Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the " dilaton ".

# The n-dimensional torus does not admit a metric with positive scalar curvature.

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

where is the Ricci tensor, is the scalar curvature,

In Riemannian geometry, the scalar curvature ( or Ricci scalar ) is the simplest curvature invariant of a Riemannian manifold.

Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface.

In general relativity, the scalar curvature is the Lagrangian density for the Einstein – Hilbert action.

this is really a deficiency in the scalar field, namely, that it is not algebraically closed.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

When data objects are stored in an array, individual objects are selected by an index that is usually a non-negative scalar integer.

The equation above is a vector equation: in a three dimensional flow, it can be expressed as three scalar equations.

For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter-clockwise rotation, and negative clockwise.

whereas if X is a complex Banach space, then the polarization identity is given by ( assuming that scalar product is linear in first argument ):

The case where X is F, and we have a bilinear form, is particularly useful ( see for example scalar product, inner product and quadratic form ).

( This is seen by writing the null vector 0 as 0 · 0 and moving the scalar 0 " outside ", in front of B, by linearity.

denotes the rank-one operator that maps the ket to the ket ( where is a scalar multiplying the vector ).

Moreover ( and more embarrassingly, although this is essentially trivial ), mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *- symbol, but an overline ( which the physicists reserve to averages ) to denote conjugate-complex numbers, i. e. for scalar products mathematicians usually write

The material derivative of any property of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body.

The curl of the gradient of any scalar field φ is always the zero vector:

If φ is a scalar valued function and F is a vector field, then

The dot product of the two vector quantities ( A and J ) is a scalar that represents the electric current.

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

In a Minkowski space, the scalar product of two four-vectors and is defined as

Then the directional derivative of is a scalar defined as

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator ().

For a collection of N point particles, the scalar moment of inertia I about the origin is defined by the equation

The scalar virial G is defined by the equation

Quintessence is a scalar field whose equation of state w < sub > q </ sub >, defined as the ratio of its pressure p < sub > q </ sub > and its density < sub > q </ sub >, is given by the potential energy and a kinetic term:

This required IEEE standard 1164, which defined the 9-value logic types: scalar < tt > std_ulogic </ tt > and its vector version < tt > std_ulogic_vector </ tt >.

We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over.

Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of W under addition and scalar multiplication, so the vector space operations are well defined.

where V is the scalar potential defined by the conservative field F.

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

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

Metre per second ( U. S. spelling: meter per second ) is an SI derived unit of both speed ( scalar ) and velocity ( vector quantity which specifies both magnitude and a specific direction ), defined by distance in metres divided by time in seconds.

Similarly, the right multiplication of a matrix A with a scalar λ is defined to be

When applied to a field ( a function defined on a multi-dimensional domain ), del may denote the gradient ( locally steepest slope ) of a scalar field ( or sometimes of a vector field, as in the Navier – Stokes equations ), the divergence of a vector field, or the curl ( rotation ) of a vector field, depending on the way it is applied.

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields ; for cartesian coordinate systems it is defined as:

A hypersurface may be locally defined implicitly as the set of points satisfying an equation, where is a given scalar function.

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

Since the curl of the electric field is zero, it is defined by a scalar electric potential field, ( see Helmholtz decomposition ).

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold ( such as a surface in space ) which takes as input a pair of tangent vectors v and w and produces a real number ( scalar ) g ( v, w ) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

Such tensor can be defined as a linear function which maps an M + N-tuple of M one-forms and N vectors to a scalar.

Recall that n-dimensional projective space is defined to be the set of equivalence classes of non-zero points in by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring are not functions on because any point has many representatives that yield different values in a polynomial ; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial.