In
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, the surface gradient is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
differential operator that is similar to the conventional
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
. The distinction is that the surface gradient takes effect along a surface.
For a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
in a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantit ...
, the surface gradient is defined and notated as
:
where
is a unit
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to the surface.
[R. Shankar Subramanian]
Boundary Conditions in Fluid Mechanics
Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface. In other words, the surface gradient is the
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal ...
of the gradient onto the surface.
The surface gradient arises whenever the gradient of a quantity over a surface is important. In the study of
capillary surface
In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid inte ...
s for example, the gradient of spatially varying
surface tension doesn't make much sense, however the surface gradient does and serves certain purposes.
See also
*
Aspect (geography)
In physical geography and physical geology, aspect (also known as exposure) is the compass direction or azimuth that a terrain surface faces.
For example, a slope landform on the eastern edge of the Rockies toward the Great Plains is desc ...
*
Geomorphometry#Surface gradient Derivatives
*
Grade (slope)
The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero ind ...
*
Spatial gradient
{{unreferenced, date=July 2016
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.
Homogeneous regions have spatial gradient ...
References
{{Reflist
Vector calculus
Surfaces
Vector physical quantities