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Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: = () , where ∇ F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation = where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: = ().
A vector field defines a direction and magnitude at each point in space. A field line is an integral curve for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line tangent to the field vector at each point.
The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. [5] [6] In particular, a vector field on can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.
The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on ...
Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise. A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point.
The curl of an order-n > 1 tensor field () is also defined using the recursive relation = ; = where c is an arbitrary constant vector and v is a vector field. Curl of a first-order tensor (vector) field
where φ is in the Sobolev space H 1 (Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl. For a slightly smoother vector field u ∈ H(curl ...