Divergence theorem commonly known as divergence theorem of gauss. This theorem also help to integrate the functions, divergence theorem is a combination of vector and integral. To understand the theorem we have to know about divergence of vector field.
Vector calculus is very important in engineering and physics to the gradient, divergence and curl. Lets take vector V(x,y,z), be a differential vector function where x y z are Cartesian coordinate and if v1,v2,v3 are component of vector V then the function div.V is called divergence of vector field. The physical meaning of divergence is value of a function that characterize a physical or geometrical property must be independent of the particular choice of coordinate that means those values must be invariant with respect to coordinate transformation.
Now divergence-theorem in integrals, this theorem is used for triple integration. Divergence-theorem, which transforms surface integral to triple integral, it also involves the divergence of vector function. The triple integration is a generalization the double integration. triple integral can be transform in to surface integral, over the boundary surface of a region in space and conversely. This is a practical interest because one of the two kinds of integral is often simpler than other. It also helps in establishing fundamental equation in fluid flow, heat conduction etc. the transformation is done by gauss divergence-theorem which include the divergence of a vector field function.
Divergence-theorem of gauss means also transformation between volume integral and surface integral. Let constant T be a closed bounded region in space whose boundary is a piece wise smooth Orientale surface and let F(x,y,z) be a vector function that is continuous and continuous first partial derivatives in some domain containing T. then we proof the divergence-theorem of gauss. In this theorem a unit normal vector of surface which is pointing to the out side of surface can also used.
Let’s clear this theorem with some Divergence Theorem Examples. Suppose we have to evaluate the surface integral by this divergence-theorem. In our problem surface S is a closed surface consisting cylindrical coordinates and also coordinates of circular disc are given. To solve this problem we need to know triple integration. Then we can solve this type of problem, when we solve this types of problem we mostly use polar coordinates, which is defined by X=rcosθ and y = rsin θ
Let’s take one more Divergence Theorem Example that is transformation of volume integral in to surface integral. Suppose function which is in vector form is given and function also involve unit normal vector. Here in place of surface volume of sphere is given, now by using gauss divergence-theorem we integrate the function from volume integral to surface integral.
