Surface Integrals of Scalar Fields: Assume is a function, is a surface S. Then the surface integral is
where the left hand side is the surface integral of the scalar field and the right hand side is the multiple integration. denotes the cross product between and ,
denotes the length of the vector
Remark. If for all , and is a surface, then
the left hand side is
the right hand side is , since and the cross product
Surface Integrals of Vector Fields:
Imagine that we have a fluid flowing through , such that determines the velocity of the fluid at . The flux is defined as the quantity of fluid flowing through per unit time.
This illustration implies that if the vector field is tangent to at each point, then the flux is zero, because the fluid just flows in parallel to , and neither in nor out. This also implies that if does not just flow along , that is, if has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of with the unit normal vector to at each point, which will give us a scalar field, and integrate the obtained field as above.
Assume is a vector field, is a surface S. Then the surface integrals of the vector field F is
The left hand side is the surface integral of vector field and the right hand side is the surface integral of scalar function, since is a scalar function. That means,
Divergence Theorem (Gauss’s theorem or Ostrogradsky’s theorem)
This theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.
where is a bounded domain and is a vector field.
where is a vector field. is a compact surface and is the boundary of The curve has the positive orientation, that means following the right hand rule.