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In the case of the gravitational field g, which can be shown to be conservative, [3] it is equal to the gradient in gravitational potential Φ: =. There are opposite signs between gravitational field and potential, because the potential gradient and field are opposite in direction: as the potential increases, the gravitational field strength decreases and vice versa.
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 ...
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.
These gradients are critical for cellular identity and cell relocation. Similarly, the gradients produced by cells may influence cellular fate by their temporal and spatial characteristics. In certain organisms, the choice of cell fate can be determined by a gradient, a binary choice, or through a relay of molecules released by a cell. [1]
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 vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] 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 ...
Biogeography now incorporates many different fields including but not limited to physical geography, geology, botany and plant biology, zoology, general biology, and modelling. A biogeographer's main focus is on how the environment and humans affect the distribution of species as well as other manifestations of Life such as species or genetic ...
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]