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The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1) k around a saddle that has k contracting dimensions and n−k expanding dimensions. The index of the vector field as a whole is defined when it has just finitely many zeroes. In ...
Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right). A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field vector at each point along its length.
In terms of the underlying linear algebra, a point belongs to a line if the inner product of the vectors representing the point and line is zero. The lines can be classified into three types. On three of the lines the binary triples for the points have the 0 in a constant position: the line 100 (containing the points 001, 010, and 011) has 0 in ...
A field is a commutative ring (F, +, *) in which 0 ≠ 1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F ×;
Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point (,) on the Euclidean plane, for any non-zero real number , the triple (,,) is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a ...
The magnetic field B can be depicted via field lines (also called flux lines) – that is, a set of curves whose direction corresponds to the direction of B, and whose areal density is proportional to the magnitude of B. Gauss's law for magnetism is equivalent to the statement that the field lines have neither a beginning nor an end: Each one ...
The principal U(1)-connection ∇ on the line bundle has a curvature F = ∇ 2, which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection d we can write ∇ = d + A and F = dA with A the 1-form composed of the electric potential and the ...
A common problem in computer graphics is to generate a non-zero vector in ℝ 3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that ...