A spatial gradient is a

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

whose

components Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...

are spatial derivatives, i.e., rate of change of a given scalar

physical quantity A physical quantity (or simply quantity) is a property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''nu ...

with respect to the position coordinates in

physical space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless fo ...

. Homogeneous regions have spatial gradient vector norm equal to zero. When evaluated over

vertical position Vertical position or vertical location is a position (mathematics), position along a vertical direction (the plumb line direction) above or below a given vertical datum (a reference level surface, such as mean sea level). Vertical distance or vert ...

(altitude or depth), it is called ''vertical derivative'' or ''vertical gradient''; the remainder is called ''horizontal gradient'' component, the

vector projection The vector projection (also known as the vector component or vector resolution) of a vector on (or onto) a nonzero vector is the orthogonal projection of onto a straight line parallel to . The projection of onto is often written as \oper ...

of the full gradient onto the

horizontal plane Horizontal may refer to: *Horizontal plane, in astronomy, geography, geometry and other sciences and contexts *Horizontal coordinate system, in astronomy *Horizontalism, in monetary circuit theory *Horizontalidad, Horizontalism, in sociology *Hor ...

. Examples: ;Biology *

Concentration gradient Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second ...

, the ratio of solute concentration between two adjoining regions * Potential gradient, the difference in electric charge between two adjoining regions ;Fluid dynamics and earth science *

Density gradient Density gradient is a spatial variation in density over a region. The term is used in the natural sciences to describe varying density of matter, but can apply to any quantity whose density can be measured. Aerodynamics In the study of supersoni ...

Pressure gradient In hydrodynamics and hydrostatics, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular locat ...

Temperature gradient A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature spatial gradient is a vector quantity with Dimensional analysis, ...

Geothermal gradient Geothermal gradient is the rate of change in temperature with respect to increasing depth in Earth's interior. As a general rule, the crust temperature rises with depth due to the heat flow from the much hotter mantle; away from tectonic plat ...

Sound speed gradient In acoustics, the sound speed gradient is the rate of change of the speed of sound with distance, for example with depth in the ocean, or height in the Earth's atmosphere. A sound speed gradient leads to refraction of sound wavefronts in the dire ...

Wind gradient In common usage, wind gradient, more specifically wind speed gradient or wind velocity gradient, or alternatively shear wind, is the vertical component of the gradient of the mean horizontal wind speed in the lower atmosphere. It is the rate of ...

Lapse rate The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude. ''Lapse rate'' arises from the word ''lapse'' (in its "becoming less" sense, not its "interruption" sense). In dry air, ...

References

* {{cite book , last=Kreyszig , first=E. , title=Advanced Engineering Mathematics , publisher=Wiley , issue=v. 10 , year=1999 , isbn=978-0-471-15496-9 , url=https://books.google.com/books?id=Xfo_AQAAIAAJ , access-date=2023-08-27

See also

References