Gradient in the context of Critical point (mathematics)

The Solar System consists of the Sun and the bodies that orbit it (most prominently Earth), being a system of masses bound together by gravity. The name comes from Sōl, the Latin name for the Sun. It formed about 4.6 billion years ago when a dense region of a molecular cloud collapsed, creating the Sun and a protoplanetary disc from which the orbiting bodies assembled. The fusion of hydrogen into helium inside the Sun's core releases energy, which is primarily emitted through its outer photosphere. This creates a decreasing temperature gradient across the system. Over 99.86% of the Solar System's mass is located within the Sun.

The most massive objects that orbit the Sun are the eight planets. Closest to the Sun in order of increasing distance are the four terrestrial planets – Mercury, Venus, Earth and Mars. These are the planets of the inner Solar System. Earth and Mars are the only planets in the Solar System which orbit within the Sun's habitable zone, where liquid water can exist on the surface. Beyond the frost line at about five astronomical units (AU), are two gas giants – Jupiter and Saturn – and two ice giants – Uranus and Neptune. These are the planets of the outer Solar System. Jupiter and Saturn possess nearly 90% of the non-stellar mass of the Solar System.

The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may be called thermo-capillary convection or Bénard–Marangoni convection.

In mathematics, the slope or gradient of a line is a number that describes the direction of the line on a plane. Often denoted by the letter m, slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same A slope is the ratio of the vertical distance (rise) to the horizontal distance (run) between two points, not a direct distance or a direct angle for any choice of points. To explain, a slope is the ratio of the vertical distance (rise) to the horizontal distance (run) between two points, not a direct distance or a direct angleThe line may be physical – as set by a road surveyor, pictorial as in a diagram of a road or roof, or abstract.An application of the mathematical concept is found in the grade or gradient in geography and civil engineering.

The steepness, incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows:

In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at the interface between two fluids with different densities. Elastic surface waves can travel along the surface of solids, such as Rayleigh or Love waves. Electromagnetic waves can also propagate as "surface waves" in that they can be guided along with a refractive index gradient or along an interface between two media having different dielectric constants. In radio transmission, a ground wave is a guided wave that propagates close to the surface of the Earth.

In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential function in n-dimensional space is typically an (n − 1)-dimensional space. The del operator illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'.

An equipotential region of a scalar potential in three-dimensional space is often an equipotential surface (or potential isosurface), but it can also be a three-dimensional mathematical solid in space. The gradient of the scalar potential (and hence also its opposite, as in the case of a vector field with an associated potential field) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.

The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the differential force of gravity, the net between gravitational forces, the derivative of gravitational potential, the gradient of gravitational fields. Therefore tidal forces are a residual force, a secondary effect of gravity, highlighting its spatial elements, making the closer near-side more attracted than the more distant far-side.

This produces a range of tidal phenomena, such as ocean tides. Earth's tides are mainly produced by the relative close gravitational field of the Moon and to a lesser extent by the stronger, but further away gravitational field of the Sun. The ocean on the side of Earth facing the Moon is being pulled by the gravity of the Moon away from Earth's crust, while on the other side of Earth there the crust is being pulled away from the ocean, resulting in Earth being stretched, bulging on both sides, and having opposite high-tides. Tidal forces viewed from Earth, that is from a rotating reference frame, appear as centripetal and centrifugal forces, but are not caused by the rotation.

In biology, a cline is a measurable gradient in a single characteristic (or biological trait) of a species across its geographical range. Clines usually have a genetic (e.g. allele frequency, blood type), or phenotypic (e.g. body size, skin pigmentation) character. They can show either smooth, continuous gradation in a character, or more abrupt changes in the trait from one geographic region to the next.

A cline is a spatial gradient in a single specific trait, rather than in a collection of traits; a single population can therefore have as many clines as it has traits, at least in principle. Additionally, as Julian Huxley recognised, these multiple independent clines may not act in concordance with each other. For example, it has been observed that in Australia, birds generally become smaller the further towards the north of the country they are found. In contrast, the intensity of their plumage colouration follows a different geographical trajectory, being most vibrant where humidity is highest and becoming less vibrant further into the arid centre of the country. Because of this, Huxley described the notion of clines as an "auxiliary taxonomic principle,” meaning that clinal variation in a species is not awarded taxonomic recognition in the way subspecies or species are.

In the context of chemistry, molecular physics, physical chemistry, and molecular modelling, a force field is a computational model that is used to describe the forces between atoms (or collections of atoms) within molecules or between molecules as well as in crystals. Force fields are a variety of interatomic potentials. More precisely, the force field refers to the functional form and parameter sets used to calculate the potential energy of a system on the atomistic level. Force fields are usually used in molecular dynamics or Monte Carlo simulations. The parameters for a chosen energy function may be derived from classical laboratory experiment data, calculations in quantum mechanics, or both. Force fields utilize the same concept as force fields in classical physics, with the main difference being that the force field parameters in chemistry describe the energy landscape on the atomistic level. From a force field, the acting forces on every particle are derived as a gradient of the potential energy with respect to the particle coordinates.

A large number of different force field types exist today (e.g. for organic molecules, ions, polymers, minerals, and metals). Depending on the material, different functional forms are usually chosen for the force fields since different types of atomistic interactions dominate the material behavior.

Iridescence (also known as goniochromism) is the phenomenon of certain surfaces that appear gradually to change colour as the angle of view or the angle of illumination changes. Iridescence is caused by wave interference of light in microstructures or thin films. Examples of iridescence include soap bubbles, feathers, butterfly wings and seashell nacre, and minerals such as opal. Pearlescence is a related effect where some or most of the reflected light is white. The term pearlescent is used to describe certain paint finishes, usually in the automotive industry, which actually produce iridescent effects.

An image gradient is a directional change in the intensity or color in an image. The gradient of the image is one of the fundamental building blocks in image processing. For example, the Canny edge detector uses image gradient for edge detection. In graphics software for digital image editing, the term gradient or color gradient is also used for a gradual blend of color which can be considered as an even gradation from low to high values, and seen from black to white in the images to the right. Another name for this is color progression.

Mathematically, the gradient of a two-variable function (here the image intensity function) at each image point is a 2D vector with the components given by the derivatives in the horizontal and vertical directions. At each image point, the gradient vector points in the direction of largest possible intensity increase, and the length of the gradient vector corresponds to the rate of change in that direction.

A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates in physical space. Homogeneous regions have spatial gradient vector norm equal to zero.When evaluated over vertical position (altitude or depth), it is called vertical derivative or vertical gradient; the remainder is called horizontal gradient component, the vector projection of the full gradient onto the horizontal plane.

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.

Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken.

A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time). Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions. Fluxions and fluents made up Newton's early calculus.