Tag Archives: Divergence 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 5: Integral theorems. Section 1: Divergence 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.

The divergence theorem is a generalisation of Green’s theorem (or more precisely of Green’s theorem in normal form). This theorem states that the flux of a vector field out of a closed surface equals to the integral of the divergence of that vector field over the volume enclosed by the surface (recall that Green’s theorem states that the flux out of a simple closed curve equals to the integral of the divergence over the region enclosed by the curve). Continue reading