Search results
Results from the WOW.Com Content Network
A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration. Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization ...
where ∇φ denotes the gradient vector field of φ. The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇φ is a conservative field.
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
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.
Field lines of a vector field v, around the boundary of an open curved surface with infinitesimal line element dl along boundary, and through its interior with dS the infinitesimal surface element and n the unit normal to the surface. Top: Circulation is the line integral of v around a closed loop C. Project v along dl, then sum.
A tensor field of order greater than one may be ... Integration around a closed curve in the clockwise sense is the negative of the same line integral in the ...
Given a vector field V and a curve γ, parametrized by t in [a, b] (where a and b are real numbers), the line integral is defined as = (()) ˙ (). To show vector field topology one can use line integral convolution .
The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.