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The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative ...
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. Examples: Biology
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The simplest definition for a potential gradient F in one dimension is the following: [1] = = where ϕ(x) is some type of scalar potential and x is displacement (not distance) in the x direction, the subscripts label two different positions x 1, x 2, and potentials at those points, ϕ 1 = ϕ(x 1), ϕ 2 = ϕ(x 2).
Individual cells within a morphogenetic field in an embryo are flexible: thus, cells in a cardiac field can be redirected via cell-to-cell signaling to replace damaged or missing cells. [6] The Imaginal disc in larvae is an example of a discrete morphogenetic field region of cells in an insect embryo. [7]
In biology, a cline is a measurable gradient in a single characteristic (or biological trait) of a species across its geographical range. [1] Clines usually have a genetic (e.g. allele frequency, blood type), or phenotypic (e.g. body size, skin pigmentation) character.
Developmental bioelectricity is a sub-discipline of biology, related to, but distinct from, neurophysiology and bioelectromagnetics. Developmental bioelectricity refers to the endogenous ion fluxes, transmembrane and transepithelial voltage gradients, and electric currents and fields produced and sustained in living cells and tissues.