Vector spherical harmonics
Vector spherical harmonics are a set of vector-valued orthogonal functions defined on the unit sphere, extending the scalar spherical harmonics to facilitate the expansion of vector fields in spherical coordinates.[1] They serve as solutions to the vector Helmholtz equation, \nabla^2 \mathbf{V} + k^2 \mathbf{V} = 0, where the components are constructed from scalar spherical harmonics Y_{\ell m}(\theta, \phi) combined with differential operators or unit vectors.[1] Typically denoted as \mathbf{Y}_{j \ell m} or similar, these functions are simultaneous eigenfunctions of the total angular momentum \mathbf{J}^2, orbital angular momentum \mathbf{L}^2, and J_z, with eigenvalues j(j+1), \ell(\ell+1), and m, respectively, where j = \ell or j = \ell \pm 1, and |m| \leq j.[2] The construction of vector spherical harmonics involves applying the angular momentum operator \mathbf{L} = -i \mathbf{r} \times \nabla to scalar harmonics or using vector couplings.[3] For the transverse electric mode (often labeled \mathbf{\Psi}_{j \ell m}), one form is \mathbf{\Psi}_{j \ell m} = \frac{1}{\sqrt{j(j+1)}} \mathbf{L} Y_{j m}, which is divergence-free and perpendicular to the radial direction.[2] Other modes include radial-longitudinal and transverse-magnetic types.[3] Key properties include orthonormality over the sphere, \int \mathbf{Y}_{j \ell}^{m*} \cdot \mathbf{Y}_{j' \ell'}^{m'} \, d\Omega = \delta_{j j'} \delta_{\ell \ell'} \delta_{m m'}, and completeness for expanding tangential vector fields, enabling separation of radial and angular dependencies in partial differential equations.[2] They also exhibit parity transformations and rotation properties inherited from scalar harmonics, with specific sets being solenoidal (\nabla \cdot \mathbf{V} = 0) or irrotational.[4] Vector spherical harmonics find extensive applications in electromagnetism for expanding radiation fields from multipole sources, as detailed in classical treatments of wave propagation and scattering by spherical objects.[3] In magnetostatics, they decompose magnetic fields from current distributions, satisfying orthogonality for efficient multipole expansions.[5] Beyond physics, they are employed in quantum mechanics for angular momentum states of particles with spin, in fluid dynamics for solving the Navier-Stokes equations around spheres (e.g., Stokes flow), and in astronomy for analyzing vector fields on the celestial sphere, such as proper motions or magnetic fields.[3][4][6]Mathematical Foundations
Scalar Spherical Harmonics
Scalar spherical harmonics, denoted Y_{l}^{m}(\theta, \phi), where l is the degree (a non-negative integer) and m is the order (an integer satisfying -l \leq m \leq l), arise as the solutions to the angular part of Laplace's equation in spherical coordinates. These functions provide a complete orthonormal basis for the space of square-integrable scalar functions on the unit sphere, enabling the expansion of any such function in terms of these harmonics. Pierre-Simon Laplace first introduced them in 1782 as part of his work on gravitational potentials, representing the coefficients in the expansion of the Newtonian potential.[7] The explicit form of the scalar spherical harmonics is given by Y_{l}^{m}(\theta, \phi) = (-1)^{m} \sqrt{ \frac{(2l+1)(l-m)!}{4\pi (l+m)!} } \, P_{l}^{m}(\cos \theta) \, e^{i m \phi}, for m \geq 0, with Y_{l}^{-m}(\theta, \phi) = (-1)^{m} (Y_{l}^{m})^{*}(\theta, \phi) for negative orders, where P_{l}^{m} are the associated Legendre functions of the first kind. The factor (-1)^{m} for positive m incorporates the Condon-Shortley phase convention, which ensures consistency in applications such as angular momentum calculations in quantum mechanics.[8][9] Key properties include their orthogonality over the sphere: \int_{0}^{2\pi} \int_{0}^{\pi} Y_{l}^{m*}(\theta, \phi) \, Y_{l'}^{m'}(\theta, \phi) \, \sin \theta \, d\theta \, d\phi = \delta_{l l'} \, \delta_{m m'}, which follows from the Sturm-Liouville theory applied to the associated Legendre equation, along with their completeness, allowing any L^{2} function on the sphere to be uniquely expanded as f(\theta, \phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} f_{l m} Y_{l}^{m}(\theta, \phi). Additionally, the addition theorem states that P_{l}(\cos \gamma) = \frac{4\pi}{2l+1} \sum_{m=-l}^{l} Y_{l}^{m*}(\theta', \phi') \, Y_{l}^{m}(\theta, \phi), where \gamma is the angle between the directions (\theta, \phi) and (\theta', \phi'), facilitating rotations and multipole expansions. In quantum mechanics, these harmonics describe the angular dependence of the hydrogen atom's wave functions, \psi_{n l m}(r, \theta, \phi) = R_{n l}(r) \, Y_{l}^{m}(\theta, \phi), where they correspond to the eigenfunctions of the angular momentum operators.[10][11]Vector Fields on the Sphere
A vector field \mathbf{V} defined on the unit sphere S^2 in three-dimensional Euclidean space is represented in spherical coordinates (\theta, \phi) as \mathbf{V} = V_r \hat{r} + V_\theta \hat{\theta} + V_\phi \hat{\phi}, where V_r, V_\theta, and V_\phi are the components along the local orthonormal basis vectors \hat{r}, \hat{\theta}, and \hat{\phi}, respectively.[12] The radial component V_r \hat{r} points normal to the sphere, while the tangential component \mathbf{V}_T = V_\theta \hat{\theta} + V_\phi \hat{\phi} lies entirely within the tangent plane at each point.[12] The basis vectors \hat{\theta} and \hat{\phi} are defined as the directions of increasing polar angle \theta and azimuthal angle \phi, with explicit Cartesian expressions \hat{\theta} = \cos\theta \cos\phi \, \hat{i} + \cos\theta \sin\phi \, \hat{j} - \sin\theta \, \hat{k} and \hat{\phi} = -\sin\phi \, \hat{i} + \cos\phi \, \hat{j}.[12] The tangential vector fields on S^2 admit an irreducible decomposition under the action of the rotation group SO(3) into two distinct parts: a polar (irrotational) component and an axial (solenoidal) component. This decomposition aligns with the Helmholtz-Hodge theorem adapted to the sphere, which uniquely expresses any tangential vector field \mathbf{V}_T as the sum of a curl-free part \nabla_s f (the surface gradient of a scalar potential f) and a divergence-free part \hat{r} \times \nabla_s [g](/page/G) (the surface curl of another scalar potential g), where \nabla_s denotes the surface gradient operator projecting the standard Euclidean gradient onto the tangent plane via the projection matrix P_s = I - \hat{r} \hat{r}^T.[12][13] The polar part transforms as a polar vector under rotations, while the axial part transforms as an axial vector, reflecting their distinct parity behaviors. For fields of fixed degree l, the space of such tangential vector fields decomposes into two irreducible SO(3)-representations, each of dimension $2l + 1, yielding a total dimension of $2(2l + 1). This structure necessitates two separate families of basis functions to span the space completely, with the components derived by applying the surface gradient and curl operators to scalar functions of degree l.[12]Definitions
Standard Definition
Vector spherical harmonics are defined for degrees l \geq 1 and orders m with |m| \leq l; they vanish for l = 0.[14] The two primary transverse families are the magnetic-type and electric-type harmonics, both tangential to the unit sphere and constructed from the scalar spherical harmonics Y_{lm}(\theta, \phi).[15] The magnetic-type vector spherical harmonics are given by \mathbf{Y}_{lm}^{(M)}(\theta, \phi) = \frac{1}{\sqrt{l(l+1)}} \mathbf{L} Y_{lm}, where \mathbf{L} = -i \mathbf{r} \times \nabla is the angular momentum operator.[14] These functions are divergence-free on the sphere and form an orthonormal basis for tangential vector fields with that property. The electric-type vector spherical harmonics are \mathbf{Y}_{lm}^{(E)}(\theta, \phi) = \frac{1}{\sqrt{l(l+1)}} \nabla_1 Y_{lm}, where \nabla_1 denotes the surface gradient operator on the unit sphere.[15] These are curl-free on the sphere (up to a factor) and complete the orthonormal basis for tangential vector fields when combined with the magnetic type. An alternative notation uses \Psi_{lm} for the magnetic type and \Phi_{lm} for the electric type. In the spherical basis \{\hat{r}, \hat{\theta}, \hat{\phi}\}, the components can be expressed explicitly; for example, the \theta-component of \Psi_{lm} is \Psi_{lm}^\theta = -\frac{m}{\sqrt{l(l+1)} \sin \theta} Y_{lm}.[4] Both transverse families satisfy the normalization condition \int |\mathbf{Y}_{lm}^{(J)}|^2 \, d\Omega = 1, \quad J = M, E, where the integral is over the unit sphere.[14] For completeness, a longitudinal type is defined as \mathbf{Y}_{lm}^{(L)}(\theta, \phi) = \hat{r} Y_{lm}(\theta, \phi), which is non-tangential and includes only a radial component.[15]Alternative Definitions
In electrodynamics, particularly for expanding solutions to Maxwell's equations in spherical coordinates, an alternative pair of vector spherical harmonics is employed to separate transverse electric (TE) and transverse magnetic (TM) modes. The magnetic-type harmonic is given by \mathbf{M}_{lm}(\theta, \phi) = -\frac{i}{\sqrt{l(l+1)}} \mathbf{r} \times \nabla Y_{lm}(\theta, \phi), while the electric-type harmonic is \mathbf{N}_{lm}(\theta, \phi) = \frac{1}{k \sqrt{l(l+1)}} \nabla \times \mathbf{L} Y_{lm}(\theta, \phi), where k is the wave number and \mathbf{L} = -i \mathbf{r} \times \nabla.[14][16] In quantum mechanics, vector spherical harmonics are often denoted as \mathbf{Y}_{J L M}(\theta, \phi), where L is the orbital angular momentum quantum number, J is the total angular momentum (J = L \pm 1 or J = L for vector coupling with spin-1), and M is the projection along the z-axis. These are constructed via angular momentum addition: \mathbf{Y}_{J L M}(\theta, \phi) = \sum_{m_L, m_S} \langle L m_L, 1 m_S | J M \rangle Y_{L m_L}(\theta, \phi) \boldsymbol{\chi}_{1 m_S}, with \boldsymbol{\chi}_{1 m_S} as the spherical basis vectors for spin-1. This notation emphasizes the coupling of orbital and intrinsic angular momentum, useful for describing particle states with definite total J. For J = L, the form is transverse like \mathbf{M}_{lm}; for J = L \pm 1, it includes longitudinal components analogous to \mathbf{N}_{lm}.[3] A geometrical approach defines vector spherical harmonics using spin-weighted spherical harmonics {}_s Y_{lm}(\theta, \phi) with spin weight s = \pm 1, which naturally describe tangent vector fields on the sphere via a local dyad basis (e.g., \mathbf{m}, \bar{\mathbf{m}} for polarization). A vector field \mathbf{F} decomposes into components F_{+1} = \mathbf{F} \cdot \mathbf{m} (spin +1) and F_{-1} = \mathbf{F} \cdot \bar{\mathbf{m}} (spin -1), expanded as F_{\pm 1} = \sum_{l m} a_{\pm 1, lm} {}_{\pm 1} Y_{lm}. These relate to standard vector harmonics through basis transformations, with {}_{\pm 1} Y_{lm} \propto (\nabla_\theta \pm i \nabla_\phi / \sin\theta) Y_{lm}, facilitating analysis of polarization and gravitational perturbations.[17] Conventions vary across literature, notably in normalization and phase factors. Jackson includes the factor of -i in \mathbf{M}_{lm} for convenience in radiation problems, ensuring real-valued fields for certain modes, while Stratton omits the i and uses \sqrt{(l \pm 1)/(2l+1)} factors in some decompositions for consistency with scalar harmonics. These differences arise from choices in orthonormal bases but preserve orthogonality integrals.[3]Properties
Orthogonality and Normalization
The inner product for vector fields on the unit sphere is defined as \langle \mathbf{U}, \mathbf{V} \rangle = \int \mathbf{U}^* \cdot \mathbf{V} \, d\Omega, where the integral extends over the solid angle d\Omega = \sin\theta \, d\theta \, d\phi. This inner product induces an L^2 Hilbert space structure on vector fields, including radial and tangential components.[18] The vector spherical harmonics \mathbf{Y}_{lm}^{(J)} satisfy orthogonality relations with respect to this inner product. For the modes J, J' = M, E, \langle \mathbf{Y}_{lm}^{(J)}, \mathbf{Y}_{l'm'}^{(J')} \rangle = \delta_{ll'} \delta_{mm'} \delta_{JJ'}, while cross terms between different modes vanish. The radial mode \mathbf{Y}_{lm}^{(L)} is orthogonal to both M and E modes and satisfies \langle \mathbf{Y}_{lm}^{(L)}, \mathbf{Y}_{l'm'}^{(L)} \rangle = \delta_{ll'} \delta_{mm'}. These properties extend the orthogonality of scalar spherical harmonics to the vector case.[19][18] The tangential vector spherical harmonics \mathbf{Y}_{lm}^{(M)} and \mathbf{Y}_{lm}^{(E)} form a complete orthonormal basis for the space of square-integrable tangential vector fields on the sphere. Any such field \mathbf{V}_T admits the expansion \mathbf{V}_T(\theta, \phi) = \sum_{l=1}^\infty \sum_{m=-l}^l \left[ a_{lm}^{(M)} \mathbf{Y}_{lm}^{(M)}(\theta, \phi) + a_{lm}^{(E)} \mathbf{Y}_{lm}^{(E)}(\theta, \phi) \right], where the coefficients are given by the projections a_{lm}^{(J)} = \langle \mathbf{Y}_{lm}^{(J)}, \mathbf{V}_T \rangle = \int \mathbf{Y}_{lm}^{(J)*} \cdot \mathbf{V}_T \, d\Omega for J = M, E. Together with the radial modes \mathbf{Y}_{lm}^{(L)}, they provide a complete basis for all square-integrable vector fields on the sphere. The radial harmonics \mathbf{Y}_{lm}^{(L)} provide a complete basis for the subspace of radial vector fields.[18][19] The normalization of the vector spherical harmonics derives from integrals involving scalar spherical harmonics. In particular, the surface gradient operator \nabla_1 on the sphere satisfies \int |\nabla_1 Y_{lm}|^2 \, d\Omega = l(l+1) \int |Y_{lm}|^2 \, d\Omega, which follows from the eigenvalue equation for the spherical Laplacian -\Delta_1 Y_{lm} = l(l+1) Y_{lm} and integration by parts. This relation ensures that the tangential harmonics \mathbf{Y}_{lm}^{(M)} and \mathbf{Y}_{lm}^{(E)}, constructed via curl and gradient of Y_{lm}, are normalized with factors such as $1/\sqrt{l(l+1)}. The radial mode \mathbf{Y}_{lm}^{(L)} is normalized directly from the scalar harmonic without the l(l+1) factor.[20][14] Orthonormality implies the Parseval identity for tangential vector fields: \int |\mathbf{V}_T|^2 \, d\Omega = \sum_{l=1}^\infty \sum_{m=-l}^l \sum_{J=M,E} |a_{lm}^{(J)}|^2. An analogous identity holds for the radial subspace.[18]Symmetry and Parity
Vector spherical harmonics transform under rotations as irreducible representations of the SO(3) group, carrying angular momentum quantum number l. Specifically, for a rotation R, the unitary representation operator U(R) acts on the basis functions as U(R) \mathbf{Y}_{l m}^{(J)} = \sum_{m'} D_{m' m}^l (R) \mathbf{Y}_{l m'}^{(J)}, where D^l_{m'm}(R) are the Wigner D-matrices and J denotes the type (electric, magnetic, or longitudinal).[5] Under parity transformation \mathbf{r} \to -\mathbf{r}, which on the unit sphere corresponds to (\theta, \phi) \to (\pi - \theta, \phi + \pi), the magnetic vector spherical harmonics \mathbf{Y}_{l m}^{(M)} are even, acquiring a phase P = +1 relative to the scalar spherical harmonics' parity (-1)^l, while the electric \mathbf{Y}_{l m}^{(E)} and longitudinal \mathbf{Y}_{l m}^{(L)} are odd with P = -1. This parity behavior links directly to multipole expansions, where even-parity modes correspond to certain radiation patterns and odd-parity to others.[21] The distinction between types arises from their vector nature: magnetic harmonics \mathbf{Y}^{(M)} behave as axial vectors (pseudovectors), transforming without an extra sign under inversion compared to polar vectors, whereas electric \mathbf{Y}^{(E)} and radial longitudinal \mathbf{Y}^{(L)} are true polar vectors. Under full spatial inversion, these properties ensure consistent classification in multipole parity, with magnetic modes preserving orientation relative to the coordinate system and electric/radial modes flipping.[5] Tangential vector spherical harmonics, both electric and magnetic, have zero radial component, ensuring they lie in the tangential plane. This orthogonality to \mathbf{\hat{r}} facilitates their use in divergence-free or curl-free decompositions.[21][5]Differential Operator Relations
Vector spherical harmonics interact with differential operators defined on the sphere through relations that reflect their construction from scalar spherical harmonics and their role in decomposing vector fields into irreducible representations under rotations. These relations are particularly useful for solving partial differential equations on spherical geometries, such as those in electromagnetism and fluid dynamics. The surface gradient operator \nabla_1, acting on a scalar spherical harmonic Y_{lm}, yields a tangential vector field proportional to the electric-type vector spherical harmonic \mathbf{Y}_{lm}^{(E)}: \nabla_1 Y_{lm} = \sqrt{l(l+1)} \, \mathbf{Y}_{lm}^{(E)}. This relation follows from the definition of \mathbf{Y}_{lm}^{(E)} as the normalized tangential gradient of the scalar harmonic, ensuring orthogonality and unit norm on the unit sphere.[22] The surface curl operator \mathrm{curl}_1, applied to a radial vector field f \hat{r}, produces a tangential vector field aligned with the magnetic-type vector spherical harmonic \mathbf{Y}_{lm}^{(M)}. Specifically, for f = Y_{lm}, \mathrm{curl}_1 (Y_{lm} \hat{r}) = \hat{r} \times \nabla_1 Y_{lm} = i \sqrt{l(l+1)} \, \mathbf{Y}_{lm}^{(M)}, up to normalization conventions that may absorb the imaginary unit for phase consistency in applications. This arises because \mathbf{Y}_{lm}^{(M)} is defined via the normalized cross product \hat{r} \times \nabla_1 Y_{lm}, making it divergence-free and solenoidal.[22] The surface divergence operator \mathrm{div}_1 highlights the properties of the tangential vector spherical harmonics. For the magnetic component, \mathrm{div}_1 \mathbf{Y}_{lm}^{(M)} = 0, reflecting its incompressibility on the sphere, whereas the electric harmonic satisfies \mathrm{div}_1 \mathbf{Y}_{lm}^{(E)} = -\sqrt{l(l+1)} \, Y_{lm}. These properties stem from the Helmholtz decomposition on the sphere, where \mathbf{Y}_{lm}^{(M)} forms the solenoidal basis, and \mathbf{Y}_{lm}^{(E)} the irrotational part. The radial mode \mathbf{Y}_{lm}^{(L)} has no tangential components, so \mathrm{div}_1 does not apply.[22] The spherical Laplacian \Delta_1 acts as an eigenvalue operator on the tangential vector spherical harmonics: \Delta_1 \mathbf{Y}_{lm}^{(J)} = -l(l+1) \, \mathbf{Y}_{lm}^{(J)}, for J = E, M, analogous to the scalar case \Delta_1 Y_{lm} = -l(l+1) Y_{lm}. This eigenvalue equation underscores their role as eigenfunctions of the vector angular momentum operator on the sphere. Extending to three dimensions, the full curl and gradient operators relate radial and tangential components. For a radial scalar field, \nabla \times (\mathbf{r} f) with f = Y_{lm} yields a vector field proportional to \mathbf{Y}_{lm}^{(M)}, specifically \nabla \times (\mathbf{r} Y_{lm}) = i r \sqrt{l(l+1)} \, \mathbf{Y}_{lm}^{(M)}, mirroring the surface curl up to radial scaling. Similarly, the divergence of a general vector field \mathbf{A} involves \nabla (\mathbf{r} \cdot \mathbf{A}), which couples to the longitudinal mode through the scalar potential. These 3D relations facilitate expansions of solutions to vector wave equations.Multipole Moments
Vector multipole moments serve as coefficients in the expansion of the vector potential generated by a localized vector source, such as a current distribution confined within a finite region. For points outside the sources, the vector potential \mathbf{A}(\mathbf{r}) admits the multipole expansion \mathbf{A}(\mathbf{r}) = \sum_{l=1}^{\infty} \sum_{m=-l}^{l} \left[ \mathbf{M}_{l m}(\hat{\mathbf{r}}) \frac{q_{l m}^{(M)}}{r^{l+1}} + \mathbf{N}_{l m}(\hat{\mathbf{r}}) \frac{q_{l m}^{(E)}}{r^{l+1}} \right], where \mathbf{M}_{l m} and \mathbf{N}_{l m} denote the toroidal and poloidal vector spherical harmonics, respectively, and the multipole moments q_{l m}^{(J)} (with J = M, E) are defined by the volume integral q_{l m}^{(J)} = \int \mathbf{j}(\mathbf{r}') \cdot \mathbf{Y}_{l m}^{(J)*} (\hat{\mathbf{r}}') \, r'^l \, dV', with \mathbf{j}(\mathbf{r}') the current density and \mathbf{Y}_{l m}^{(J)} the appropriate vector spherical harmonics conjugate to \mathbf{M}_{l m} or \mathbf{N}_{l m}. The electric (polar) multipole moments q_{l m}^{(E)} originate from charge distributions and the longitudinal components of the current, contributing to the poloidal part of the field, whereas the magnetic (toroidal) moments q_{l m}^{(M)} arise from transverse currents or equivalent magnetic sources, driving the toroidal field structure. In the context of far-field radiation for time-harmonic sources, the static radial dependence $1/r^{l+1} generalizes to outgoing spherical waves via spherical Hankel functions of the first kind, h_l^{(1)}(kr), which incorporate phase factors e^{ikr} and amplitude scaling with kr for large kr \gg 1, enabling the description of radiating multipoles. For the dipole case l=1, these vector multipole moments reduce to the familiar scalar electric and magnetic dipole moments, bridging the vector expansion to the standard scalar multipole theory. This expansion is unique, owing to the completeness and orthogonality of the vector spherical harmonics, and it converges for all r larger than the radial extent of the sources.Explicit Forms and Examples
Construction Methods
Vector spherical harmonics are typically constructed from scalar spherical harmonics by applying tangential components of the gradient and curl operators, yielding the electric (poloidal) and magnetic (toroidal) modes, respectively. The electric-type vector spherical harmonic is expressed in spherical coordinates as \mathbf{Y}_{l m}^{(E)}(\theta, \phi) = \frac{1}{\sqrt{l(l+1)}} \left( \hat{\theta} \frac{\partial Y_{l m}}{\partial \theta} + \hat{\phi} \frac{i m Y_{l m}}{\sin \theta} \right), where Y_{l m} denotes the scalar spherical harmonic. This form arises from the tangential gradient of the scalar harmonic, normalized to ensure unit norm over the sphere. The magnetic-type counterpart is obtained via azimuthal derivatives, given by \mathbf{Y}_{l m}^{(M)}(\theta, \phi) = \frac{1}{\sqrt{l(l+1)}} \left( \hat{\theta} \frac{i m Y_{l m}}{\sin \theta} - \hat{\phi} \frac{\partial Y_{l m}}{\partial \theta} \right), corresponding to the tangential curl and ensuring orthogonality to the electric mode. Recursive relations facilitate computation across magnetic quantum numbers m. Raising and lowering operators J_{\pm} act on the scalar Y_{l m} as J_{\pm} Y_{l m} = \hbar \sqrt{(l \mp m)(l \pm m + 1)} Y_{l, m \pm 1}, after which the vector construction is applied to generate \mathbf{Y}_{l, m \pm 1}^{(E/M)}.[23] These ladder operators preserve the vector harmonic properties under rotation.[23] Integral representations employ addition theorems, expressing vector spherical harmonics in terms of zonal harmonics for special cases like m=0. For instance, the addition theorem for vector modes links them to scalar addition formulas via tensor products, enabling efficient evaluation in multipole expansions.[24] Numerical implementation relies on fast recursion for associated Legendre functions underlying Y_{l m}, using three-term relations like (l - m + 1) P_l^m(x) = (2l - 1) x P_{l-1}^m(x) - (l + m - 1) P_{l-2}^m(x) to compute values stably from the equator toward the poles, avoiding singularities at \theta = 0, \pi.[25] Derivatives for the vector components are then evaluated analytically from these. As of 2025, libraries such as Mathematica provide built-in construction viaSphericalHarmonicY combined with vector operators, while Python's SciPy integrates with packages like Windspharm for vector spherical harmonic transforms and computations.[26][27]