How do you evaluate the divergence theorem?

Example 1 Use the divergence theorem to evaluate ∬S→F⋅d→S ∬ S F → ⋅ d S → where →F=xy→i−12y2→j+z→k F → = x y i → − 1 2 y 2 j → + z k → and the surface consists of the three surfaces, z=4−3×2−3y2 z = 4 − 3 x 2 − 3 y 2 , 1≤z≤4 1 ≤ z ≤ 4 on the top, x2+y2=1 x 2 + y 2 = 1 , 0≤z≤1 0 ≤ z ≤ 1 on the sides and z=0 on the …

What is the divergence theorem formula?

∭ E div F d V = ∬ S F · d S . ∭ E div F d V = ∬ S F · d S . Figure 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.

How do you prove the Gauss divergence theorem?

Divergence Theorem Proof The divergence theorem-proof is given as follows: Assume that “S” be a closed surface and any line drawn parallel to coordinate axes cut S in almost two points. Let S1 and S2 be the surface at the top and bottom of S. These are represented by z=f(x,y)and z=ϕ(x,y) respectively.

When can you apply the divergence theorem?

Surface must be closed But unlike, say, Stokes’ theorem, the divergence theorem only applies to closed surfaces, meaning surfaces without a boundary. For example, a hemisphere is not a closed surface, it has a circle as its boundary, so you cannot apply the divergence theorem.

What is divergence theorem examples?

Example 1. and S is surface of box 0≤x≤1,0≤y≤3,0≤z≤2. Use outward normal n. We compute the triple integral of divF=3+2y+x over the box B: ∬SF⋅dS=∫10∫30∫20(3+2y+x)dzdydx=∫10∫30(6+4y+2x)dydx=∫10(18+18+6x)dx=36+3=39.

How do you prove the Gauss theorem?

When an area of the surface projected in a plane perpendicular to the field is multiplied by the electric field is the electric flux. It states that, the total electric flux of a given surface is equal to the 1Eθ times of the total charge enclosed in it or amount of charge contained within that surface.

What does Gauss theorem state?

Gauss’s law for electricity states that the electric flux across any closed surface is proportional to the net electric charge enclosed by the surface. The law implies that isolated electric charges exist and that like charges repel one another while unlike charges attract.

When can you apply divergence theorem?

What is divergence theorem explain with example?

The theorem states that the outward flux of a vector field through a closed surface is equal to the triple integral of the divergence of the vector field inside the surface. …