Monthly Archives: May 2010

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

Advertisements

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: Rectifiable parameterised curves

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 discuss curves and their lengths in more detail. Let n\in\mathbb{N} be an arbitrary natural number which is fixed throughout this post. Note that in general one needs to distinguish between a parameterised curve, which is a continuous mapping \boldsymbol{c}:[a,b]\to\mathbb{R}^n, and the curve \mathcal{C}, which is the image of \boldsymbol{c} given by \{\boldsymbol{c}(t) \in \mathbb{R}^n: t\in [a,b]\}. Here we shall discuss parameterised curves. Hence, for instance, the parameterised curve \boldsymbol{c}(t)= \cos t \widehat{\boldsymbol{i}} + \sin t \widehat{\boldsymbol{j}} with 0 \le t \le 4\pi is a circle traversed twice and has therefore length 4\pi, whereas its image is just a circle which has length 2\pi. 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: Recommended reading: Line integrals and orientation

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 discuss in what sense the scalar line integrals do not depend on the orientation of the curve.

Let us start with integrals \int_a^b f(t)\,\mathrm{d} t where f is a Riemann integrable function. Let a \textless b and let P_n=\{x_0,\ldots, x_n\} be such that a=x_0 \textless x_1 \textless x_2 \textless \cdots \textless x_{n-1} \textless x_n =b, let x_{i-1} \le t_i \le x_{i} for 1\le i \le n and \delta(P_n)=\max_{1\le i\le n} x_{i+1}-x_i. We can now form the Riemann sum

\displaystyle \mathcal{S}(P_n) = \sum_{i=1}^n f(t_i) (x_i-x_{i-1}).

By considering the limit \lim \mathcal{S}(P_n) where n\to\infty such that \delta(P_n) \to 0, we obtain the Riemann integral \int_a^b f(t)\,\mathrm{d} t. 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

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