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the surface gradient is a vector differential operator that is similar to the conventional gradient. The distinction is that the surface gradient takes
Surface_gradient
Multivariate derivative (mathematics)
y, z). The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level
Gradient
Gradient whose components are spatial derivatives
Image gradient Time derivative Material derivative Structure tensor Surface gradient Kreyszig, E. (1999). Advanced Engineering Mathematics. Wiley. ISBN 978-0-471-15496-9
Spatial_gradient
Rate of increase in wind strength per unit increase in height
In common usage, wind gradient, more specifically wind speed gradient or wind velocity gradient, or alternatively shear wind, is the vertical component
Wind_gradient
Science of using a material's refractive index for optical effects
flat surfaces, or lenses that do not have the aberrations typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that
Gradient-index_optics
Concept in fluid dynamics
of a liquid due to gradients in the surface tension of the liquid. Diffusion is of whatever is creating the gradient in the surface tension. Thus as the
Marangoni_number
Analysis of terrain
disadvantages. The surface normal at any point on the terrain surface is a vector ray that is perpendicular to the surface. The surface gradient (∇f) is the
Geomorphometry
Optimization algorithm
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate
Gradient_descent
Force resulting from a difference in pressure across a surface
fluid mechanics, the pressure-gradient force is the force that results when there is a difference in pressure across a surface. In general, a pressure is
Pressure-gradient_force
Temperature difference per unit of length
The absorption of solar light at or near the planetary surface increases the temperature gradient and results in convection (a major process of cloud formation
Temperature_gradient
Rate of temperature increase with depth in Earth's interior
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
Geothermal_gradient
Angle to the horizontal plane
The grade (US) or gradient (UK) (also called slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the
Grade_(slope)
Compass direction that a slope faces
is complemented by grade (or inclination angle) to characterize the surface gradient. Aspect can have a strong influence on temperature. This is because
Aspect_(geography)
Line or vector perpendicular to a curve or a surface
surface is given by the gradient 𝐧=∇F(x,y,z). since the gradient at any point is perpendicular to the level set of S. For a surface S in ℝ3 given as the
Normal_(geometry)
Physical phenomenon between two fluids
regions of low surface tension. The surface tension gradient can be caused by concentration gradient or by a temperature gradient (surface tension is an
Marangoni_effect
surface, where the flow is driven by a concentration gradient in the solution. This is distinct from flow relative to a surface driven by a gradient in
Diffusiophoresis and diffusioosmosis
Diffusiophoresis_and_diffusioosmosis
Physical phenomenon
"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
Surface_wave
Evaluates a line integral through a gradient field using the original scalar field
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated
Gradient_theorem
Space rate of variation of pressure in a given direction
gradient is a two-dimensional vector resulting from the projection of the pressure gradient onto a local horizontal plane. Near the Earth's surface,
Pressure_gradient
Term in road design
Drainage gradient (DG) is a term in road design, defined as the combined slope due to road surface cross slope (CS) and longitudinal slope (hilliness)
Drainage_gradient
Elliptic partial differential equation
difference between V and the gradient of f. In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary
Poisson's_equation
Unsorted glacial sediment
flow tills vary, and can depend on factors such as water content, surface gradient, and debris characteristics. Generally, flow tills with a higher water
Till
Mathematical descriptions of molecular diffusion
substrate surface and a near-surface concentration gradient is never formed; It can also be used to predict the adsorption rate of molecules on the surface when
Fick's_laws_of_diffusion
Vehicle dynamics terms
if the understeer gradient is positive, oversteer if the understeer gradient is negative, and neutral steer if the understeer gradient is zero. Car and
Understeer_and_oversteer
Type of gradient noise in computer graphics
Perlin noise is a type of gradient noise developed by Ken Perlin in 1983. Its application include procedurally generating terrain, applying pseudo-random
Perlin_noise
Rate of change of speed of sound with distance
inversely proportional to the gradient. When the sun warms the Earth's surface, there is a negative temperature gradient in atmosphere. The speed of sound
Sound_speed_gradient
Variation of the redox potential with distance (or depth)
with an oxidizing environment at the surface and increasingly reducing conditions below the surface. Redox gradients are generally understood at the macro
Redox_gradient
Integration over a non-flat region in 3D space
particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double
Surface_integral
Detachment of a boundary layer from a surface into a wake
gradient d u o / d s ( s ) < 0 {\displaystyle du_{o}/ds(s)<0} along the surface, which in turn is directly related to the pressure and its gradient by
Flow_separation
Computer vision framework
Gradient vector flow (GVF), a computer vision framework introduced by Chenyang Xu and Jerry L. Prince, is the vector field that is produced by a process
Gradient_vector_flow
Surface slope along a watercourse
Stream gradient (or stream slope) is the grade (or slope) of a stream. It is measured by the ratio of drop in elevation and horizontal distance. It is
Stream_gradient
Mathematical identities
function f(x,y,z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad(f)=∇f=(∂∂x, ∂∂y, ∂∂z)f=∂f∂x𝐢+∂f∂y𝐣+∂f∂z𝐤
Vector_calculus_identities
Matrix of partial derivatives of a vector-valued function
scalar-valued function of several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivative
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Operation in calculus
general manifolds (curves, surfaces, and their higher-dimensional analogs). The exterior derivative plays the role of the gradient and curl of vector calculus
Integral
Aspects of fluid mechanics involving fluid flow
describe a layer of relatively inviscid fluid with a free surface, in which surface gradients are small. Darcy's law is used for flow in porous media,
Fluid_dynamics
Optimization algorithm for artificial neural networks
In machine learning, backpropagation is a gradient computation method commonly used for training a neural network in computing parameter updates. It is
Backpropagation
Theorem in calculus
through a closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of
Divergence_theorem
Third planet from the Sun
being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all of Earth's water is contained in its ocean, which covers
Earth
Oil spread from ships to calm waves
and as waves locally stretch or compress, it leads to a concentration gradient inducing tangential shear forces leading to extra dissipation and damping
Storm_oil
Derivative of a function with multiple variables
∂f∂xn(a)). This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f
Partial_derivative
measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional
Willmore_energy
Pressure gradient in which pressure increases in the direction of fluid flow
along the surface and hence the lift and drag characteristics. Turbulent boundary layers tend to be able to sustain an adverse pressure gradient better than
Adverse_pressure_gradient
Lowest part of the atmosphere directly influenced by contact with the planetary surface
affected by surface drag and turns across the isobars (see Ekman layer for more detail). Typically, due to aerodynamic drag, there is a wind gradient in the
Planetary_boundary_layer
Natural movement of air or other gases relative to a planet's surface
windsurfing. In gliding, wind gradients just above the surface affect the takeoff and landing phases of flight of a glider. Wind gradient can have a noticeable
Wind
Layer where the density gradient is greatest within a body of water
A pycnocline is the cline or layer where the density gradient (∂ρ/∂z) is greatest within a body of water. An ocean current is generated by the forces
Pycnocline
Surface representing the interface between two different fluids
\gamma } is the surface tension associated with the interface, and ∇ S {\displaystyle \scriptstyle \nabla _{S}} is the surface gradient. Note that the
Capillary_surface
Fluid surface of constant density
isopycnic surface is a surface of constant density inside a fluid. Isopycnic surfaces contrast with isobaric or isothermal surfaces, which describe surfaces of
Isopycnic
Vector operator in vector calculus
{G} ).} The Laplacian of a scalar field is the divergence of the field's gradient: div ( grad φ ) = Δ φ . {\displaystyle \operatorname {div} (\operatorname
Divergence
Theorem in vector calculus
the behavior of a vector field along the edge of a surface to the behavior of its curl on the surface itself. In its usual three-dimensional form, it says
Stokes'_theorem
Distinct layer of temperature change in a body of water
water, as in an ocean or lake; or air, e.g. an atmosphere) with a high gradient of distinct temperature differences associated with depth. In the ocean
Thermocline
Shoreline formations made up of various grades of sediment in an arc pattern
cusps is caused by the communication of surface gradients along the beach by the smoothing of the beach surface as the beach tries to rearrange itself
Beach_cusps
Biological membrane that separates the interior of a cell from its outside environment
and is propelled by the concentration gradient created by each side of the membrane. Such a concentration gradient across a semipermeable membrane sets
Cell_membrane
Region in space where every point is at the same potential
an equipotential surface (or potential isosurface), but it can also be a three-dimensional mathematical solid in space. The gradient of the scalar potential
Equipotential
Measure of electric field through surface
crosses a given surface. The electric flux through a closed surface is directly proportional to the total charge contained within that surface. The electric
Electric_flux
Sustainable energy from sea and river water
Osmotic power, salinity gradient power or blue energy is the energy available from the difference in the salt concentration between seawater and river
Osmotic_power
Matrix of second derivatives
matrix of a function f is the transpose of the Jacobian matrix of the gradient of the function f; that is: 𝐇(f(𝐱))=𝐉(∇f(𝐱))T. If f is a homogeneous
Hessian_matrix
Differentiation under the integral sign formula
derivative of the vector field, Σ is a surface bounded by the closed curve ∂Σ, dA is a vector element of the surface Σ, ds is a vector element of the curve
Leibniz_integral_rule
Process of water movement through a plant
the hydraulic conductivity of the soil and the magnitude of the pressure gradient through the soil. Both of these factors influence the rate of bulk flow
Transpiration
Instantaneous rate of change of the function
x, and one has ∇𝐯f(𝐱)=∇f(𝐱)⋅𝐯 where the ∇ on the right denotes the gradient and ⋅ is the dot product. It can be derived by using the property that
Directional_derivative
Vector difference of geostrophic wind movement at high and low altitudes
unit vector, and the subscript "p" on the gradient operator denotes gradient on a constant pressure surface) with respect to pressure, and integrate from
Thermal_wind
Unit of gravitational gradient
so the sum of gravity gradient along any three perpendicular directions is 0. Near the surface of earth, the equipotential surface is nearly spherical,
Eotvos_(unit)
Underwater diving breathing gas supplied from the surface
Surface-supplied diving is a mode of underwater diving using equipment supplied with breathing gas through a diver's umbilical from the surface, either
Surface-supplied_diving
Curve along which a 3D surface is at equal elevation
hypothetical surface with one or more horizontal planes. The configuration of these contours allows map readers to infer the relative gradient of a parameter
Contour_line
Difference in wind speed or direction over a short distance
shear (/ʃɪər/; also written windshear), sometimes referred to as wind gradient, is a difference in wind speed and/or direction over a relatively short
Wind_shear
Type of weather map
at the leading edge of a sharp temperature gradient on an isotherm analysis, often marked by a sharp surface pressure trough. Cold fronts can move up to
Surface_weather_analysis
Four-vector analogue of the gradient operation
geometry, the four-gradient (or 4-gradient) ∂ {\displaystyle {\boldsymbol {\partial }}} is the four-vector analogue of the gradient ∇ → {\displaystyle
Four-gradient
Instantaneous rate of change (mathematics)
real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable f(x) is differentiable at a point
Derivative
Physical quantity
fluids or Euler equations (fluid dynamics) for ideal inviscid fluid, the gradient of pressure becomes a function of body forces only. The Navier-Stokes momentum
Hydrostatic_pressure
State of balance between external forces on a fluid and internal pressure gradient
as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing
Hydrostatic_equilibrium
Technique in analytical chemistry
phase to reduce the surface tension of water. Gradient elution uses this effect by automatically reducing the polarity and the surface tension of the aqueous
High-performance liquid chromatography
High-performance_liquid_chromatography
Notation of differential calculus
that the operator ∇ will also be treated as an ordinary vector. ∇φ Gradient: The gradient gradφ of the scalar field φ is a vector, which is symbolically expressed
Notation_for_differentiation
Anti-reflective surface
surfaces in the form of cones produces highly water-repellent surfaces. These nano-cone textures are superhydrophobic or super-water-hating. Gradient
Nanotextured_surface
In linguistics, conformity with grammar
whether a particular word, phrase, or clause is well-formed. This problem of gradient well-formedness, uncertainty about the well-formedness of a particular
Well-formedness
Formula for the derivative of a ratio of functions
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Quotient_rule
Study of mathematical algorithms for optimization problems
only (sub)gradient information and others of which require the evaluation of Hessians. Methods that evaluate gradients, or approximate gradients in some
Mathematical_optimization
Differential calculus on function spaces
suggests that if we can find a function ψ {\displaystyle \psi } whose gradient is given by P , {\displaystyle P,} then the integral A {\displaystyle A}
Calculus_of_variations
with respect to an adjacent charged surface. This force might be electric, pressure gradient, concentration gradient, or gravity. In addition, the moving
Electrokinetic_phenomena
Ramp from elevated sidewalk to road
inches (10–15 cm) above the street surface) a curb cut is finished at an intermediate gradient that connects both surfaces, sometimes with tactile paving
Curb_cut
Measure of misalignment between the gradients of pressure and density in a fluid
these two surfaces is not horizontal and the system is close to hydrostatic equilibrium, the gradient of the pressure is vertical but the gradient of the
Baroclinity
the surface toward minimal energy by a gradient descent method. The aim can be to find a minimal energy surface, or to model the process of evolution by
Surface_Evolver
Statistical approach
Box–Behnken design Central composite design Gradient-enhanced kriging (GEK) IOSO method based on response-surface methodology Optimal designs Plackett–Burman
Response_surface_methodology
Study of rates of change
analogous methods use the gradient and the Hessian matrix. A critical point of a scalar-valued function occurs where the gradient is zero, and the eigenvalues
Differential_calculus
Model of atmospheric motion
to allow for different surface roughness.) In the cross-stream momentum equation, the Coriolis force and normal pressure gradient are both negligible, leading
Balanced_flow
Optimization algorithm
differs from gradient descent methods, which adjust all of the values in x {\displaystyle \mathbf {x} } at each iteration according to the gradient of the hill
Hill_climbing
When potential energy difference depends only on displacement
that fluid can be obtained by integrating the negative pressure gradient along the surface of the object: F B = − ∮ S ∇ p ⋅ d S . {\displaystyle F_{B}=-\oint
Scalar_potential
Deviation from the normal change of an atmospheric property with altitude
conditions, the normal vertical temperature gradient is inverted so that the air is colder near the surface of the Earth. This can occur when, for example
Inversion_(meteorology)
Global climate phenomenon
east–west temperature gradient and reduces the strength of the overturning. This, in turn, reduces the easterly winds at the ocean surface, reduces the upwelling
El_Niño–Southern_Oscillation
Corrective lens used in eyeglasses
the bottom of the lens. The length of the progressive power gradient on the lens surface depends on the design of the lens, with a final addition power
Progressive_lens
Vertical rate of change of temperature in atmosphere
extremely small lapse rate, the Tolman gradient (derived by R. C. Tolman in 1930). At Earth's surface, the Tolman gradient would be about Γt=Ts×(10−16m−1),
Lapse_rate
Concept in hydrology
height of the water surface in the tube relative to a common datum. The hydraulic head can be used to determine a hydraulic gradient between two or more
Hydraulic_head
Algorithm for finding zeros of functions
Newton's method can be used for solving optimization problems by setting the gradient to zero. Arthur Cayley in 1879 in The Newton–Fourier imaginary problem
Newton's_method
Directional mass flow of oceanic water
circulation that is driven by global density gradients created by surface heat and freshwater fluxes. Wind-driven surface currents (such as the Gulf Stream) travel
Ocean_current
Differential operator in mathematics
or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols
Laplace_operator
in concentration (emergence of concentration gradients in the solution adjacent to the electrode surface) is the difference in the rate of electrochemical
Concentration_polarization
Body of salt water covering most of Earth
towards the surface. Deep ocean water has a temperature between −2 °C (28 °F) and 5 °C (41 °F) in all parts of the globe. The temperature gradient over the
Ocean
Meteorological concept
the influence from the surface and the boundary layer is negligible. Although the framework is formulated with the gradients of free tropospheric temperature
Weak temperature gradient approximation
Weak_temperature_gradient_approximation
Body of surface water flowing down a channel
A stream is a continuous body of surface water flowing within the bed and banks of a channel. The channel is a watercourse, although the watercourse may
Stream
Concept in atmospheric science
reflects the horizontal pressure gradient as the driving force of atmospheric motion, while the actual near-surface wind requires the synergistic modulation
Geostrophic_wind
Weather or lab measurements
Antarctic where surface temperatures drop to approximately −98 °C (−144 °F; 175 K). Due to the very strong temperature gradient near the surface, these imply
Lowest temperature recorded on Earth
Lowest_temperature_recorded_on_Earth
Absorption of liquid by a colloid
in volume. Water surface potential movement takes place along a concentration gradient; some dry materials absorb water. A gradient between the absorbent
Imbibition
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