# Tag Archives: Several Variable Calculus

## 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

## 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

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

## Math2111: Chapter 4: Surface integrals. Section 3: Surface integrals of vector fields

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 Section 2 we have seen how to calculate surface area and integrate scalar fields over surfaces. We now establish an extension similar to the step from scalar line integrals to vector line integrals.

We assume again that the surfaces we consider are smooth, that is, they are assumed to be images of parameterised surfaces $\boldsymbol{\Phi}:D\to\mathbb{R}^3$ for which:

• $D$ is a non-empty, compact and Jordan-measurable subset of $\mathbb{R}^2$;
• the mapping $\boldsymbol{\Phi}$ is one-to-one;
• $\boldsymbol{\Phi}$ is continuously differentiable;
• the normal vector $\boldsymbol{n}=\frac{\partial \boldsymbol{\Phi}}{\partial u} \times \frac{\partial \boldsymbol{\Phi}}{\partial v} \neq \boldsymbol{0}$ except possibly at a finite number of points;

(Notice, the condition that ${}D$ is compact can also be replaced by the condition that the surface $S=\{\boldsymbol{\Phi}(u,v): (u,v)\in D\}$ is compact.)

## Math2111: Chapter 4: Surface integrals. Section 2: Surface area and surface integrals of scalar fields

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 the following we assume that the surfaces are smooth, that is, they are assumed to be images of parameterised surfaces $\boldsymbol{\Phi}:D\to\mathbb{R}^3$ for which:

• $D$ is a non-empty, compact and Jordan-measurable subset of $\mathbb{R}^2$;
• the mapping $\boldsymbol{\Phi}$ is one-to-one;
• $\boldsymbol{\Phi}$ is continuously differentiable
• the normal vector $\boldsymbol{n}=\frac{\partial \boldsymbol{\Phi}}{\partial u} \times \frac{\partial \boldsymbol{\Phi}}{\partial v} \neq \boldsymbol{0}$ except possibly at a finite number of points;

(Notice, the condition that ${}D$ is compact can also be replaced by the condition that the surface $S=\{\boldsymbol{\Phi}(u,v): (u,v)\in D\}$ is compact.)

## Math2111: Chapter 4: Surface integrals. Section 1: Parameterisations of surfaces

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.

Previously we have seen parameterised curves. We now generalise this idea to surfaces.

Definition
Let $D\subseteq \mathbb{R}^2$ be Jordan measurable. By a parameterised surface we mean a continuously differentiable function $\boldsymbol{\Phi}:D\to\mathbb{R}^3$. The image $\boldsymbol{\Phi}(D)$ is called surface.