Verify the divergence theorem for vector field F ( x, y, z ) = 〈 x + y + z, y, 2 x − y 〉 F ( x, y, z ) = 〈 x + y + z, y, 2 x − y 〉 and surface S given by the cylinder x 2 + y 2 = 1, 0 ≤ z ≤ 3 x 2 + y 2 = 1, 0 ≤ z ≤ 3 plus the circular top and bottom of the cylinder. As the volumes of the approximating boxes shrink to zero, this approximation becomes arbitrarily close to the flux over S. When adding up all the fluxes, the only flux integrals that survive are the integrals over the faces approximating the boundary of E.
If an approximating box shares a face with another approximating box, then the flux over one face is the negative of the flux over the shared face of the adjacent box. Just as in the informal proof of Stokes’ theorem, adding these fluxes over all the boxes results in the cancelation of a lot of the terms. On the other hand, the sum of div F Δ V div F Δ V over all the small boxes approximating E is the sum of the fluxes over all these boxes. The sum of div F Δ V div F Δ V over all the small boxes approximating E is approximately ∭ E div F d V. This approximation becomes arbitrarily close to the value of the total flux as the volume of the box shrinks to zero. Total flux ≈ ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) Δ V = div F Δ V. The area of the top of the box (and the bottom of the box) Δ S Δ S is Δ x Δ y. The dot product of F = 〈 P, Q, R 〉 F = 〈 P, Q, R 〉 with k is R and the dot product with − k − k is − R. The normal vector out of the top of the box is k and the normal vector out of the bottom of the box is − k. Let the center of B have coordinates ( x, y, z ) ( x, y, z ) and suppose the edge lengths are Δ x, Δ y, Δ x, Δ y, and Δ z Δ z ( Figure 6.88(b)). Let B be a small box with sides parallel to the coordinate planes inside E ( Figure 6.88). This explanation follows the informal explanation given for why Stokes’ theorem is true.
#ELECTRIC FLUX EQUATION CLOSED SURFACE FULL#
However, we look at an informal proof that gives a general feel for why the theorem is true, but does not prove the theorem with full rigor. The proof of the divergence theorem is beyond the scope of this text. Therefore, the divergence theorem is a version of Green’s theorem in one higher dimension. Recall that the flux form of Green’s theorem states that ∬ D div F d A = ∫ C F Overview of Theoremsīefore examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed:įigure 6.87 The divergence theorem relates a flux integral across a closed surface S to a triple integral over solid E enclosed by the surface. We use the theorem to calculate flux integrals and apply it to electrostatic fields. The divergence theorem has many uses in physics in particular, the divergence theorem is used in the field of partial differential equations to derive equations modeling heat flow and conservation of mass.
In this section, we state the divergence theorem, which is the final theorem of this type that we will study. We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain.
6.8.3 Apply the divergence theorem to an electrostatic field.6.8.2 Use the divergence theorem to calculate the flux of a vector field.6.8.1 Explain the meaning of the divergence theorem.
0 kommentar(er)
