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The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation. The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field.
Isoclines are often used as a graphical method of solving ordinary differential equations. In an equation of the form y' = f(x, y), the isoclines are lines in the (x, y) plane obtained by setting f(x, y) equal to a constant. This gives a series of lines (for different constants) along which the solution curves have the same gradient.
For a flow, the vector field v(x) is an affine function of the position in the phase space, that is, ˙ = = +, with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity).
It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution. More precisely, the system has the following form x ˙ = ε f ( x , t , ε ) , 0 ≤ ε ≪ 1 {\displaystyle {\dot {x}}=\varepsilon f(x,t,\varepsilon ...
The Cremona diagram, also known as the Cremona-Maxwell method, is a graphical method used in statics of trusses to determine the forces in members (graphic statics). The method was developed by the Italian mathematician Luigi Cremona .
This equation says that the vector tangent to the curve at any point x(t) along the curve is precisely the vector F(x(t)), and so the curve x(t) is tangent at each point to the vector field F. If a given vector field is Lipschitz continuous, then the Picard–Lindelöf theorem implies that there exists a unique flow for small time.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .