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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.
Vector field : Vector field plots (or quiver plots) show the direction and the strength of a vector associated with a 2D or 3D points. They are typically used to show the strength of the gradient over the plane or a surface area. Violin plot : Violin plots are a method of plotting numeric data.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
This is a list of graphical methods with a mathematical basis. Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization . There is also a list of computer graphics and descriptive geometry topics .
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 .
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation = where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: = ().
In a broad sense, the term graphic statics is used to describe the technique of solving particular practical problems of statics using graphical means. [1] Actively used in the architecture of the 19th century, the methods of graphic statics were largely abandoned in the second half of the 20th century, primarily due to widespread use of frame structures of steel and reinforced concrete that ...
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.