Tag Archives: Stokes 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 2: 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 now generalise Green’s theorem to vector fields in \mathbb{R}^3. The following result is by Sir George Gabriel Stokes.

Stokes’ theorem
Let \mathcal{C} be a simple closed positively oriented curve in the plane \mathbb{R}^2 parameterised by a continuously differentiable function \boldsymbol{c}:[a,b]\to\mathbb{R}^2 and let {}D be the region enclosed by \mathcal{C}. Let \boldsymbol{\Phi}:D\to\mathbb{R}^3, \boldsymbol{\Phi}=\boldsymbol{\Phi}(u,v), be a twice continuously differentiable and one-to-one parameterisation of a surface S=\mbox{Image}(\boldsymbol{\Phi}). The boundary \partial S of {}S is parameterised by the positively oriented curve \boldsymbol{\Phi}(\boldsymbol{c}):[a,b]\to\mathbb{R}^3.

Then

\iint_D (\nabla \times \boldsymbol{F})(\boldsymbol{\Phi}(u,v)) \cdot \left(\frac{\partial \boldsymbol{\Phi}}{\partial u} \times \frac{\partial \boldsymbol{\Phi}}{\partial v}\right)(u,v) \,  \mathrm{d} u \, \mathrm{d} v = \int_{\partial S} \boldsymbol{F}\cdot \mathrm{d} \boldsymbol{s}.

Continue reading