Gauss Divergence Theorem Quiz

Gauss Divergence Theorem

• The theorem states that the surface integral of the normal component of a vector A taken over a closed surface is equal to the integral of the divergence of A taken over the volume enclosed by the surface.

• The formula for the theorem is:

  ∫∫(A • n̂)dS = ∫∫∫ (∇ • A)dV

• Where:

  • A is a vector function of position with continuous derivatives
  • n̂ is the positive (outward drawn) normal to the surface S
  • V is the volume bounded by a closed surface S
  • ∇ • A is the divergence of A

Applying the Theorem

• When applying the Gauss Divergence Theorem, remember to pay attention to signs and ensure that P is multiplied by x and Q is multiplied by y.

• The theorem can be applied to surfaces represented by z = f(x,y), x = g(y,z), or y = h(x,z), where f,g,h are single-valued, continuous, and differentiable.

• The position vector to any point of S is:

  r = xi + yj + f(x,y)k

• The vector r is perpendicular to n, the normal vector.

Simplifying the Formula

• The formula can be further simplified to:

  ∫∫(A • n̂)dS = ∫∫F(x,y)dxdy

• Where R is the projection of S on the xy plane and F(x,y) is the value of A at a point on S.