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This function represents half of the rate of energy dissipation of the system through friction. The force of friction is negative the velocity gradient of the dissipation function, F → f = − ∇ v R ( v ) {\displaystyle {\vec {F}}_{f}=-\nabla _{v}R(v)} , analogous to a force being equal to the negative position gradient of a potential.
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).
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...
Potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field , which is a valid approximation for several applications.
As with a negative corona, a positive corona is initiated by an exogenous ionization event in a region of a high potential gradient. The electrons resulting from the ionization are attracted toward the curved electrode, and the positive ions repelled from it. By undergoing inelastic collisions closer and closer to the curved electrode, further ...
The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature.
Figure FHN: To mimick the action potential, the FitzHugh–Nagumo model and its relatives use a function g(V) with negative differential resistance (a negative slope on the I vs. V plot). For comparison, a normal resistor would have a positive slope, by Ohm's law I = GV, where the conductance G is the inverse of resistance G=1/R.
The pixels with the largest gradient values in the direction of the gradient become edge pixels, and edges may be traced in the direction perpendicular to the gradient direction. One example of an edge detection algorithm that uses gradients is the Canny edge detector. Image gradients can also be used for robust feature and texture matching.