Integral Theorems


Gauss’s Divergence Theorem:

Statement. It states that the volume integral of the divergence of a vector  field A, taken over  any volume, V is equal to the surface integral of A taken over the closed surface surrounding  the volume V and vice versa.

Stoke’s Theorem:

Statement. It states  that the surface integral of curl of a vector field over an open surface equals the line integral of the vector field over the closed curve bounding the surface area and vice versa.

Helmholtz’s Theorem:

Statement. It states that a vector field is completely specified  by itsdivergence and curl or in other words any vector field may be expressed as the sum of an irrotational vector  and a solenoidal vector.

Significance of Helmholtz Theorem:

To study electromagnetic fields, we need to specify electric and magnetic field vectors at a space point at a  given time uniquely. Helmholtz theorem suggests that  each of these field vectors can be uniquely specified  by assigning its divergence and curl at the point of interest at a given time. There are two divergence and two curl equations governing the electromagnetic theory. These four  equations commonly known as Maxwell’s equations, are used for the study of electromagnetics. So, these equations are also known as electromagnetic field equations.

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