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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.
No magnetic field in the direction of propagation. These are sometimes called E modes because there is only an electric field along the direction of propagation. Hybrid modes Non-zero electric and magnetic fields in the direction of propagation. See also Planar transmission line § Modes.
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 ...
The circulation Γ of a vector field V around a closed curve C is the line integral: [3] [4] =. In a conservative vector field this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken.
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: = automatically results in the identity ...
A frame is set on a specific spacetime point, not an extending field or a flux line as a mathematical object. It is a different issue if you consider flux as a physical entity (see Magnetic flux quantum), or consider the effective/relative definition of motion/rotation of a field (see below). This note helps resolve the paradox.
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 curve (top) is filled according to two rules: the even-odd rule (left), and the non-zero winding rule (right). In each case an arrow shows a ray from a point P heading out of the curve. In the even-odd case, the ray is intersected by two lines, an even number; therefore P is concluded to be 'outside' the curve.