Dimension (physics) in the context of "Gravitational field"

Geopotential height, also known as geopotential altitude or geopotential elevation, is a vertical coordinate (with dimension of length) representing the work involved in lifting one unit of mass over one unit of length through a hypothetical space in which the acceleration of gravity is assumed constant. Geopotential heights are referenced to Earth's mean sea level, taking its best-fitting equigeopotential as a reference surface or vertical datum.In SI units, a geopotential height difference of one meter implies the vertical transport of a parcel of one kilogram; adopting the standard gravity value (9.80665 m/s), it corresponds to a constant work or potential energy difference of 9.80665 joules.

Geopotential height differs from geometric height (as given by a tape measure) because Earth's gravity is not constant, varying markedly with altitude and latitude; thus, a 1-m geopotential height difference implies a different vertical distance in physical space: "the unit-mass must be lifted higher at the equator than at the pole, if the same amount of work is to be performed".It is a useful concept in meteorology, climatology, and oceanography; it also remains a historical convention in aeronautics as the altitude used for calibration of aircraft barometric altimeters.

A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number. Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve.In physical space, a 1D subspace is called a "linear dimension" (rectilinear or curvilinear), with units of length (e.g., metre).

In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field ${\displaystyle K}$ is a one-dimensional vector space over itself. The projective line over ${\displaystyle K,}$ denoted ${\displaystyle \mathbf {P} ^{1}(K),}$ is a one-dimensional space. In particular, if the field is the complex numbers ${\displaystyle \mathbb {C} ,}$ then the complex projective line ${\displaystyle \mathbf {P} ^{1}(\mathbb {C} )}$ is one-dimensional with respect to ${\displaystyle \mathbb {C} }$ (but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates).

In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m) or pascal (Pa).

Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface (such as a piston) push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Stress is frequently represented by a lowercase Greek letter sigma (σ).

In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division) or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense and ${\displaystyle 6+{\tfrac {2}{3}}=6.66...}$ (a repeating decimal) in the second sense.

In metrology (International System of Quantities and the International System of Units), "quotient" refers to the general case with respect to the units of measurement of physical quantities. Ratios is the special case for dimensionless quotients of two quantities of the same kind.Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration (e.g., "per second"), are known as rates.For example, density (mass divided by volume, in units of kg/m) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio". Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".

A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

Strain has dimension of a length ratio, with SI base units of meter per meter (m/m).Hence strains are dimensionless and are usually expressed as a decimal fraction or a percentage.Parts-per notation is also used, e.g., parts per million or parts per billion (sometimes called "microstrains" and "nanostrains", respectively), corresponding to μm/m and nm/m.