Green's Theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green.

Let C be a positively oriented, piecewise smooth, simple closed curve in the plane ², and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol.

In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.

(From Wikipedia)

Vector Differentiation

In calculus we compute derivatives of real functions of a real variable. In the case of functions of a single variable

            y = f(x)

we compute the derivative of y with respect to x. In the case of a function of several variables

            y = f(x₁, x₂, ... , x_n)

we can compute the derivative of y with respect to any one of the independent variables, viewing the others as fixed.

In vector analysis we compute derivatives of vector functions of a real variable; that is we compute derivatives of functions of the type

            F(t) = f₁(t) i + f₂(t) j + f₃(t) k

This function can be viewed as describing a space curve. Intuitively it can be regarded as a position vector, expressed as a function of t, that traces out a space curve with increasing values of t. Expressed in different notation it is the function
      R(t) = x(t) i + y(t) j + z(t) k