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The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0. 2. ...

A vector field is a map f:R^n|->R^n that assigns each x a vector f(x). Several vector fields are illustrated above. A vector field is uniquely specified by giving its ...

The field axioms are generally written in additive and multiplicative pairs. name addition multiplication associativity (a+b)+c=a+(b+c) (ab)c=a(bc) commutativity a+b=b+a ...

A divergenceless field can be partitioned into a toroidal and a poloidal part. This separation is important in geo- and heliophysics, and in particular in dynamo theory and ...

A vector field X on a compact foliated manifold (M,F) is nice if X is transverse to F and if X has a closed orbit C (called a nice orbit) such that the intersection C ...

A solenoidal vector field satisfies del ·B=0 (1) for every vector B, where del ·B is the divergence. If this condition is satisfied, there exists a vector A, known as the ...

A compact set W_infty with area mu(W_infty)=8/9(24)/(25)(48)/(49)...=pi/4 created by punching a square hole of length 1/3 in the center of a square. In each of the eight ...

The degree (or relative degree, or index) of an extension field K/F, denoted [K:F], is the dimension of K as a vector space over F, i.e., [K:F]=dim_FK. If [K:F] is finite, ...

The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński ...

The vector field N_f(z)=-(f(z))/(f^'(z)) arising in the definition of the Newtonian graph of a complex univariate polynomial f (Smale 1985, Shub et al. 1988, Kozen and ...