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There does not exist an everywhere nonzero tangent vector field on the 2-sphere S^2. This implies that somewhere on the surface of the Earth, there is a point with zero ...

Defined for a vector field A by (A·del ), where del is the gradient operator. Applied in arbitrary orthogonal three-dimensional coordinates to a vector field B, the ...

Let V be a vector space over a field K, and let A be a nonempty set. Now define addition p+a in A for any vector a in V and element p in A subject to the conditions: 1. ...

The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) ...

Let E be a linear space over a field K. Then the vector space tensor product tensor _(lambda=1)^(k)E is called a tensor space of degree k. More specifically, a tensor space ...

The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

Newton's method for finding roots of a complex polynomial f entails iterating the function z-[f(z)/f^'(z)], which can be viewed as applying the Euler backward method with ...

A vector space V with a ring structure and a vector norm such that for all v,W in V, ||vw||<=||v||||w||. If V has a multiplicative identity 1, it is also required that ...

The curl of a vector field, denoted curl(F) or del xF (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at ...

The term used in physics and engineering for a harmonic function. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a ...