A stream function for F = 〈 P, Q 〉 F = 〈 P, Q 〉 is a function g such that P = g y P = g y and Q = − g x. There is a stream function g ( x, y ) g ( x, y ) for F.In other words, flux is independent of path. If C 1 C 1 and C 2 C 2 are curves in the domain of F with the same starting points and endpoints, then ∫ C 1 F.N d s across any closed curve C is zero.The following statements are all equivalent ways of defining a source-free field F = 〈 P, Q 〉 F = 〈 P, Q 〉 on a simply connected domain (note the similarities with properties of conservative vector fields): If we replace “circulation of F” with “flux of F,” then we get a definition of a source-free vector field. In fact, if the domain of F is simply connected, then F is conservative if and only if the circulation of F around any closed curve is zero. Recall that if vector field F is conservative, then F does no work around closed curves-that is, the circulation of F around a closed curve is zero. ∫ C v ⋅ N ds = ∬ D ( P x + Q y ) d A = ∬ D 8 d A = 8 ( area of D ) = 80. Therefore, by the same logic as in Example 6.40, Let D be any region with a boundary that is a simple closed curve C oriented counterclockwise. The logic of the previous example can be extended to derive a formula for the area of any region D. In Example 6.40, we used vector field F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 to find the area of any ellipse. Therefore, the area of the ellipse is π a b. d r = 1 2 ∫ C − y d x + x d y = 1 2 ∫ 0 2 π − b sin t ( − a sin t ) + a ( cos t ) b cos t d t = 1 2 ∫ 0 2 π a b cos 2 t + a b sin 2 t d t = 1 2 ∫ 0 2 π a b d t = π a b.r 4 ( t ) d t = ∫ a b P ( t, c ) d t + ∫ c d Q ( b, t ) d t − ∫ a b P ( t, d ) d t − ∫ c d Q ( a, t ) d t = ∫ a b ( P ( t, c ) − P ( t, d ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t = − ∫ a b ( P ( t, d ) − P ( t, c ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t.Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa. This form of the theorem relates the vector line integral over a simple, closed plane curve C to a double integral over the region enclosed by C. The first form of Green’s theorem that we examine is the circulation form. In particular, Green’s theorem connects a double integral over region D to a line integral around the boundary of D. Green’s theorem also says we can calculate a line integral over a simple closed curve C based solely on information about the region that C encloses. Green’s theorem says that we can calculate a double integral over region D based solely on information about the boundary of D. Green’s theorem takes this idea and extends it to calculating double integrals. Figure 6.32 The Fundamental Theorem of Calculus says that the integral over line segment depends only on the values of the antiderivative at the endpoints of. If the electric field is uniform, the electric flux passing through a surface of vector area S is For simplicity in calculations, it is often convenient to consider a surface perpendicular to the flux lines. Electric flux is proportional to the total number of electric field lines going through a surface. The density of these lines corresponds to the electric field strength, which could also be called the electric flux density: the number of "lines" per unit area. Note that field lines are a graphic illustration of field strength and direction and have no physical meaning. In pictorial form, this electric field is shown as a dot, the charge, radiating "lines of flux". The electric field is the gradient of the potential.Īn electric charge, such as a single electron in space, has an electric field surrounding it. The electric field E can exert a force on an electric charge at any point in space. In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow.
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