solenoidal vector वाक्य
उदाहरण वाक्य
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- For example, every solenoidal vector field can be written as
- Thus solenoidal vector fields are precisely those that have volume-preserving flows.
- The solenoidal vector fields are those with div " X " = 0.
- It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field.
- It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field.
- 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.
- 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.
- Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative ( divergence ) vanishes, and is called an incompressible flow ( sometimes solenoidal vector field ).
- One important property of the-field produced this way is that magnetic-field lines neither start nor end ( mathematically, is a solenoidal vector field ); a field line either extends to infinity or wraps around to form a closed curve.
- This is mathematically equivalent to saying that the divergence of is zero . ( Such vector fields are called solenoidal vector fields . ) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.
- Because an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field ( satisfying appropriate smoothness and decay conditions ) can be decomposed as the sum of the form-\ operatorname { grad } \ Phi + \ operatorname { curl } \ mathbf { A } where is a scalar field, called scalar potential, and is a vector field called a vector potential.
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