🧠Scalar and Vector Potentials: Decomposing Vector Fields and Their Properties

This section delves into the fundamental ways vector fields can be expressed as derivatives of other fields—either as the negative gradient of a scalar potential or the curl of a vector potential. It explores the inherent properties of such fields (curl-free for scalar potentials, divergence-free for vector potentials), their connection to conservative fields, methods for constructing these potentials, and the crucial concept of their non-uniqueness. Finally, it introduces the Helmholtz Decomposition Theorem, which states that any vector field can be uniquely decomposed into a curl-free and a divergence-free component, each derivable from a respective potential.

🎬Animated result and interactive web

The line integral result of a dot moving along a straight and a curved path between the points is independent of the path

The gravitational potential is the scalar potential of the gravitational field

The work of a conservative force field is path-independent

how integration constants behave during partial integration

how knowing one partial derivative allows us to find the form of the potential and how the other partial derivative is used to resolve the unknown function of integration

how the method for deriving the electric potential from a constant electric field extends from two dimensions to an arbitrary number of dimensions

how a scalar potential generates a curl-free and conservative vector field

how a vector potential generates a divergence-free vector field

why vector fields derived from scalar potentials are always curl-free

divergence free vector fields have vector potentials

the non-uniqueness of the vector potential and the concept of gauge transformation

any vector field that can be expressed as the curl of another vector field must necessarily be divergence-free

how the curl of different vector potentials can lead to the same result

visualize the constant vector field and the two different vector potentials

An incompressible fluid flow and the existence and non-uniqueness of its vector potential

how a general vector field can be decomposed into its curl-free and divergence-free components

gauge freedom for a divergence-free vector field showing how different vector potentials can describe the same physical field

the constant electric field's scalar potential as horizontal equipotential planes and how the negative gradient of these planes yields the original electric field

how vectors combine and interact in 3D space to produce scalar or vector results

Check the components of a vector can be found through the scalar product with the basis vectors

Computing magnitude and inner product and cross product and angles and The volume of the parallelepiped spanned of three vectors

Computing cross product of the vectors through symbolic components

how intermediate vectors are formed

the cross product of two vectors is orthogonal to both

sequentially prove that the vectors are normalized and orthogonal and display the corresponding transformation coefficients

the sum of the diagonal elements of the Kronecker delta tensor equals the dimensionality of the space

The Cross Product and the Right-Hand Rule