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Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves , space curves , polyhedra , ordinary differential equations , partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films ...
For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have
An example of a signal-flow graph Flow graph for three simultaneous equations. The edges incident on each node are colored differently just for emphasis. An example of a flow graph connected to some starting equations is presented. The set of equations should be consistent and linearly independent. An example of such a set is: [2]
A system of equations is a set of simultaneous equations, usually in several unknowns for which the common solutions are sought. Thus, a solution to the system is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. [1] It is said that commutative diagrams play the role in category theory that equations play in ...
In mathematics, a diagram algebra is an algebraic structure in which operations are performed using diagrams rather than traditional techniques. In particular, diagrammatic equations can be constructed and manipulated. Usually such algebras have a traditional counterpart which is categorically equivalent.
An animated cobweb diagram of the logistic map = (), showing chaotic behaviour for most values of >. A cobweb plot , known also as Lémeray Diagram or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions , such as the logistic map .
For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves.