Vector Analysis: Summation and Operators

Flashcards covering key theorems and operators in vector analysis, including Helmholtz's theorem, irrotational and solenoidal vectors, and the Laplacian operator.

Helmholtz's Theorem

Any vector field that is continuously differentiable in some volume V can be uniquely determined if its divergence and curl are known throughout the volume and its value is known on the surface S that bounds the volume.

Irrotational Vector

A vector for which the curl (\nabla \times \mathbf{A}) is zero throughout a region; it can be represented as the a gradient of a scalar field (\mathbf{A} = \nabla f).

Solenoidal Vector

A vector for which the divergence (\nabla \cdot \mathbf{A}) is zero throughout a region; it can be represented as the curl of a vector field (\mathbf{A} = \nabla \times \mathbf{G}).

Vector Field Decomposition

Any continuously differentiable vector field in some region can be expressed as the sum of an irrotational vector and a solenoidal vector (\mathbf{A} = \nabla f + \nabla \times \mathbf{G}).

Laplacian Operator (\nabla^2)

A combined operation where a gradient operation is followed by a divergence operation (\nabla \cdot \nabla), applicable to both scalar and vector fields.

Vector Field Uniqueness

For most vectors in electromagnetics, they can be uniquely specified when their curl and divergences are known at all points in space, assuming surface integrals vanish over all space.