Tag Archives: Green’s theorem

Math2111: Chapter 5: Recommended reading: The fundamental theorems

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.

In this entry we show that all the fundamental theorems (fundamental theorem of calculus, fundamental theorem of line integrals, Green’s theorem, Stokes’ theorem and the divergence theorem) are based on the same principle. Further, we will see that those theorems are all the fundamental theorems in \mathbb{R}, \mathbb{R}^2 and \mathbb{R}^3. Continue reading

Math2111: Chapter 5: Additional Material: Differential forms and the general Stokes’ theorem

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.

We have studied a variety of generalisations of the fundamental theorem of calculus:

Now we show how all of these formulae concisely fit into one approach. Differential forms provide the underlying theory to present all formulae in one framework. This also allows us to generalise the theorems above to arbitrary dimensions. Continue reading

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

Math2111: Chapter 3: Line integrals. Section 4: Green’s theorem

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.

We now establish another generalisation of the fundamental theorem of calculus which is known as Green’s theorem (named after George Green).

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.

Green’s theorem in normal form Continue reading