# Tag Archives: Green’s theorem in tangential form

## Math2111: Chapter 3: Additional Material: For which curves and regions does Green’s theorem apply?

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

Here we discuss the conditions under which Green’s theorem (see also here) applies, more precisely for which curves and regions enclosed by the curve.

The formula in Green’s theorem states that

$\displaystyle \oint_{\mathcal{C}} \boldsymbol{F} \cdot \mathrm{d} \boldsymbol{s} = \iint_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm{d} A, \qquad\qquad\qquad (1)$

where $\boldsymbol{F}:\mathbb{R}^2\to\mathbb{R}^2,$ $\boldsymbol{F}=P\widehat{\boldsymbol{i}}+Q\widehat{\boldsymbol{j}},$ is a continuously differentiable vector field, $\mathcal{C}$ is a curve which is simple, closed, smooth and oriented positively, and ${}R$ is the region enclosed by the curve $\mathcal{C}.$ Continue reading

In Section 3 we established the Fundamental theorem of line integrals which gives a relationship between line integrals of conservative vector fields and the values of the potential function at the endpoints of the curve. Green’s theorem on the other hand gives a relationship between double integrals and line integrals in $\mathbb{R}^2.$