Differential (infinitesimal) in the context of "Inexact differential"

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation respectively. The curl of a field is formally defined as the circulation density at each point of the field.

A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential ${\displaystyle dQ}$ for some differentiable function ${\displaystyle Q}$ in an orthogonal coordinate system (hence ${\displaystyle Q}$ is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).

An exact differential is sometimes also called a total differential, or a full differential, or, in the study of differential geometry, it is termed an exact form.

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as ${\displaystyle W=\mathbf {F} \cdot \mathbf {s} }$, have natural continuous analogues in terms of line integrals, in this case ${\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }$, which computes the work done on an object moving through an electric or gravitational field F along a path ${\displaystyle L}$.