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In physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux .
By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just ...
Through the superposition principle, given a linear ordinary differential equation (ODE), =, one can first solve =, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x 1, ..., x n) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.
which is a linear wave equation for the velocity potential φ. Again the oscillatory part of the velocity vector v is related to the velocity potential by v = ∇ φ , while as before Δ is the Laplace operator , and c is the average speed of sound in the homogeneous medium .
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
Laplace's equation is linear, and is one of the most elementary partial differential equations. This simple equation yields the entire solution for both V and p because of the constraint of irrotationality and incompressibility. Having obtained the solution for V and p, the consistency of the pressure gradient with the accelerations can be noted.
Taken together with the velocity potential, the stream function may be used to derive a complex potential. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.