Week 9: Closure

Continue working on the the preparatory exercises and the in-class exercises that you have not yet completed.

Key Concepts

Long Day will cover:

• Parametrizations of curves in \(\mathbb{R}^n\)

• Curve length

• The line integral

• The universal Jacobian (function)

• Vector fields and gradient vector fields

• The tangental line integral: Integration of vector field along a curve

• The antiderivative problem in \(\mathbb{R}^n\)

• The stair-line method

• Flow curves

If there are still concepts you are unsure about, you should reread the relevant chapters in the textbook or revisit the exercises of the week.

Extra Exercises

This week there are no closure exercises. If you have completed all the exercises of the week, go back to the exercises from previous weeks if you are still missing some there. Otherwise, spend the time on preparing for your topic in the project period.

Below follows a couple of exercises that are not part of the course syllabus but which may be relevant for certain students in the project period.

Note

The topics on surface integrals, flux, and divergence are not part of the course syllabus. These topics are treated in the following non-mandatory exercises.

1: Surface Area of a Sphere

Consider a sphere in \(\mathbb{R}^3\) centered at \((0,0,0)\) with radius \(a > 0\). Consider the boundary of this sphere:

\[\begin{equation*} \{ \pmb{x} \in \mathbb{R}^3 \mid \Vert \pmb{x} \Vert = a \}. \end{equation*}\]

The surface area of such sphere is (as is known from elementary geometry) \(4 \pi a^2\). Derive this formula by using a surface integral along with a parametrization of the sphere.

2: Flux through Parameter Surface

We are given the vector field

\[\begin{equation*} \pmb{V}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \pmb{V}(x,y,z)=(\cos(x),\cos(x)+\cos(z),0) \end{equation*}\]

along with a surface \(\mathcal{F}\) given by the parametric representation

\[\begin{equation*} \pmb{r}(u,v)=(u,0,v), \quad u\in\left[ 0,\pi\right] ,\quad v\in\left[ 0,2\right]. \end{equation*}\]

Question a

Determine the normal vector \(\pmb{n}_{\mathcal{F}}(u,v)\) that correponds to this parametrization. Argue that the parametrization is regular. Then calculate the flux of the vector field through the surface.

Answer

\[\begin{equation*} \int_{\mathcal{F}} \pmb{V} \cdot \mathrm{d} \pmb{S} = -\sin(2)\pi \end{equation*}\]

Question b

What is the significance of the sign of the flux? Can you change the sign of the flux by remaking the surface parametrization?

Question c

We are given a vector field

\[\begin{equation*} \pmb{V}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \pmb{V}(x,y,z)=(yz,-xz,x^2+y^2) \end{equation*}\]

and a surface \(\mathcal{F}\) given by the parametric representation

\[\begin{equation*} \pmb{r}(u,v)=(u\sin(v),-u\cos( v),uv), \quad u\in\left[ 0,1\right] ,\quad v\in\left[ 0,1\right]. \end{equation*}\]

Determine the normal vector \(\pmb{n}_{\mathcal{F}}(u,v)\) that corresponds to the parametrization. Argue that the parametrization is regular. Calculate the flux of the vector field through the surface.

Answer

\[\begin{equation*} \int_{\mathcal{F}} \pmb{V} \cdot \mathrm{d} \pmb{S} = \frac{3}{8} \end{equation*}\]

3: The Coulomb Vector Field

Coulomb (1736-1806) worked with electromagnetism. From his work we know the so-called Coulomb vector field:

\[\begin{equation*} \pmb{V}: \mathbb{R}^3\setminus \{(0,0,0)\} \to \mathbb{R}^3, \quad \pmb{V}(x,y,z)= \left(x^2+y^2+z^2\right)^{-\frac32} \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \end{equation*}\]

Note that the Coulomb vector field cannot be defined on all of \(\mathbb{R}^3\). But the set \(U = \mathbb{R}^3\setminus \{(0,0,0)\}\) is open, which is a default assumption for vector field domains in the textbook.

A massive cylinder \(B\) of height \(2h\) and diameter \(2a\), where \(a\) and \(h\) are positive real numbers, is given by the parametric representation

\[\begin{equation*} \pmb{r}(u,v,w)=\left(u\cos(w),u\sin(w),v\right), \quad u\in\left[0,a\right], \; v\in[-h,h], \; w\in \left[-\pi,\pi\right]. \end{equation*}\]

Question a

Draw a sketch of \(B\) (with pencil and paper, but you are welcome to try with a Sympy plot as well), and provide a parametric representation for each of the three boundary pieces that the boundary \(\partial B\) of \(B\) consists of: the bottom, the top, and the pipe-shaped, cylindrical part.

Hint

As we are working with surfaces, each parametrization should use two parameters.

Question b

Determine the flux of \(\pmb{V}\) out through \(\partial B\),

\[\begin{equation*} \int_{\partial B} \pmb{V} \cdot \mathrm{d} \pmb{S}, \end{equation*}\]

by computing the flux through each of the three pieces that \(\partial B\) consists of. How does the size of the cylinder influence the strength of the flux? In continuation hereof: What is the limit value of the flux strength as \(a\) and \(h\) go towards \(0\)?

Hint

Remember to check the directions of your normal vectors. They must all point away from \(B\).

4: Flux via the Divergence Theorem

The Divergens Theorem (also called Gauss’ Divergence Theorem) states the following:

---

Theorem (Divergence): Let \(\pmb{V}\) be a \(C^1\) vector field on an open set \(U\subseteq \mathbb{R}^3\), and let \(B \subseteq U\) be a bounded subset with a piecewise \(C^1\) boundary \(\mathcal{F}=\partial B\). Suppose \(\pmb{r}: \Gamma \to \mathbb{R}^3\), \(\Gamma \subset \mathbb{R}^2\), is a parametrization of the surface \(\mathcal{F}\) with outward-pointing normal. Then

(2)\[\begin{equation} \int_{\partial B} \pmb{V} \cdot \mathrm{d} \pmb{S} =\int_{B}\mathrm{div} (\pmb{V}) \, \mathrm{d} X. \end{equation}\]

---

The divergence \(\mathrm{div} (\pmb{V})\) is defined as the trace of the Jacobian matrix:

(3)\[\begin{equation} \mathrm{div} (\pmb{V}) = \mathrm{tr} (\pmb{J}_{\pmb{V}}). \end{equation}\]

If we think of the vector field as a velocity field of a fluid, then the divergence is a measure of the infinitesimal expansion/contraction rate of the fluid. The velocity field of an incompressible fluid has zero divergence.

We are given the \(C^1\) vector field

\[\begin{equation*} \pmb{V}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \pmb{V}(x,y,z)=(-8x,8,4z^3) \end{equation*}\]

and a spatial region

\[\begin{equation*} \Omega=\lbrace (x,y,z)\,\vert\, x^2+y^2+z^2\leq a^2\,\, \mathrm{og}\,\, z\geq 0\rbrace\,,\,a>0, \end{equation*}\]

whose surface \(\,\partial \Omega\,\) is oriented with an outwards-pointing unit normal vector field \(\,\pmb n_{\,\partial \Omega}\,\).

Question a

Calculate the volume integral

\[\begin{equation*} \int_{\Omega}\mathrm{div}(\pmb{V})\, \mathrm{d} X. \end{equation*}\]

Hint

\(\Omega\) can be parametrized by

\[\begin{equation*} \pmb{r}(u,v,w)=(u\sin v \cos w,u\sin v\sin w,u\cos v), \end{equation*}\]

\[\begin{equation*} u\in\left[ 0,a\right] ,\, v\in\left[ 0,\frac{\pi}{2}\right] ,\, w\in\left[ -\pi,\pi\right]\,. \end{equation*}\]

Question b

Calculate the surface integral of the vector field:

\[\begin{equation*} \int_{\partial\,\Omega}\,\pmb{V} \cdot \mathrm{d} \pmb{S}. \end{equation*}\]

Hint

Note that the surface of \(\Omega\) consists of two parts: a semi-spherical shell and a circular bottom surface.

Answer

If Gauss was right then you should get the same result in both questions. The answer is

\[\begin{equation*} 8a^3\pi\,(\frac 15 a^2 - \frac 23)\,. \end{equation*}\]

Question c

For which \(\,a\,\) is the flux positive with the given normal vector orientation (meaning “the outwards flow through \(\partial \Omega\) is larger than the inflow”).

Answer

The flux is negative for \(\,0<a<\frac{\sqrt{30}}{3}\,\), otherwise positive.

Question d

Can Gauss’ Divergence Theorem about the relation between the divergence integral and the surface integral of the vector field be considered as a generalization of the fundamental theorem of calculus:

\[\begin{equation*} \left[ F(x)\right] _a^b=\int_a^b F'(x)\mathrm{d}x\,? \end{equation*}\]

Answer

The divergence can be considered as a fitting replacement for the derivative of a function \(F'(x)\). In both cases we can say that we have pushed the integration ‘’to the boundary’’: For an interval \([a,b]\), the boundary is the end-points \(a\) and \(b\), while for a volume integral over a region it is the region’s surface.