Where and are solenoidal and irrotational projection operators satisfying 1 } } and and are the non-conservative and conservative parts of the body force.
12.
The Fundamental Theorem of Vector Calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.
13.
Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.
14.
Due to this last property, the solutions for the Navier Stokes equations are searched in the set of solenoidal ( " divergence-free " ) functions.
15.
Since \ psi \ boldsymbol { \ nabla } \ chi is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.
16.
In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
17.
Where the transverse current "'J " "'t " is the solenoidal current ( see Helmholtz decomposition ) generated by a second particle.
18.
A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
19.
The particle motion occurs along the figure-8 trajectory in yz plane set perpendicular to the solenoid axis with arbitrary azimuth angle \ varphi due to axial symmetry of the solenoidal magnetic field.
20.
In electromagnetism the "'B "'lines form solenoidal rings around the source electric current, whereas in aerodynamics, the air currents form solenoidal rings around the source vortex axis.
