Uge 9: Exercises

Exercises – Long Day

Note

By the Jacobian, we will (at DTU) refer to \(\sqrt{\det(\pmb{J}^T \pmb{J})}\) (i.e., the “distortion factor”, meaning the area/volume correction factor), which more officially is known as the geometric tensor. The term Jacobian is generally also used for both the Jacobian matrix as well as the Jacobian determinant (the determinant of the Jacobian matrix), so one should be cautious when using this term and be careful with how it is defined in context. In the exercises we will thus consistently use the term Jacobian function for \(\sqrt{\det(\pmb{J}^T \pmb{J})}\).

1: Line Integral of a Scalar Function. By Hand

We consider a circle \(\mathcal{C}\) in \((x,y,z)\) space given by

\[\begin{equation*} \mathcal{C}=\left\{(x,y,z)\in \mathbb{R}^3 \mid x^2+(y-1)^2=4 \wedge z=1\right\}. \end{equation*}\]

Question a

State the center and radius of \(\mathcal{C}\). Choose a parametric representation \(\pmb r(u)\) for \(\mathcal{C}\) corresponding to one complete revolution of the circle. Determine its corresponding Jacobian function.

Answer

Center is at \((0,1,1)\). A possible choice of parametrization is \(\pmb r(u)=(2\cos(u),2\sin(u)+1,1)\) where \(u \in [0,2\pi[\,\). The Jacobian function is \(\Vert \pmb r'(u) \Vert =2\).

Question b

We are given the function \(f(x,y,z)=x^2+y^2+z^2\). Determine the restriction \(f(\pmb r(u))\), and calculate the line integral

\[\begin{equation*} \int_\mathcal{C} f(x,y,z)\mathrm{d}s. \end{equation*}\]

Answer

The restriction of the function to the curve (using the parametrization chosen in question a is \(f(\pmb r(u))=6+4\sin(u)\). The line integral of \(f\) along \(\mathcal{C}\) is:

\[\begin{equation*} \int_\mathcal{C} f(x,y,z)\mathrm{d}s = 24 \pi. \end{equation*}\]

Question c

In the textbook, it is mentioned that the line integral is independent of the chosen parameterization of the circle. Try with other parameterizations, and compute the line integral based on them. To get a new parametrization of the same geometry, you can, for instance, change the direction of traversal or the traversal speed, or you can re-parameterize, for example using \(t = -2\pi u\) with a fitting new parameter interval.

Question d

Does the line integral depend on the location of the circle? For example, try translating/moving the circle by 1 unit in the direction of the \(y\) axis and compute the line integral again.

Answer

The line integral of course depends on the location of the circle. This is to be expected when the function \(f\) is not constant.

2: The Length of a Hanging Cable

We consider a non-elastic, freely hanging cable suspended between two poles, influenced only by gravity (and the tension forces in the cable). The mathematical form describing the shape of this cable is known as a catenary. The equation is

\[\begin{equation*} y = a \cosh (x/a), \end{equation*}\]

where \(a\) is the distance to the lowest point above the \(x\) axis. We will assume that the cable is fixed at \(y=5\) (meaning, \(y \in [a,5]\)).

Question a

Assume that \(0 < a \le 5\). Provide a parametrization for the curve

\[\begin{equation*} \mathcal{C}_a = \{ (x,y) \in \mathbb{R}^2 \mid y = a \cosh (x/a) \, \wedge \, y \le 5\}. \end{equation*}\]

In particular, pay attention to your choice of parameter interval (you may use SymPy’s solve command). Then, find the norm of the tangent vector (that is, the Jacobian function).

Question b

Plot the curve for \(a=0.5, a=1, a=2\). Write out the integration formula for the length of the curve \(\mathcal{C}_a\). Find a decimal approximation of the length of the curves \(\mathcal{C}_{0.5}\), \(\mathcal{C}_1\) and \(\mathcal{C}_2\).

3: Line Integral of Vector Field I. By Hand

In the \((x,y)\) plane we are given a vector field

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

as well as a curve \(\mathcal{C}\) by the equation

\[\begin{equation*} y=x^2, \quad x\in\left[ -1,1\right]. \end{equation*}\]

Question a

Provide a parametrization of \(\mathcal{C}\). Determine the corresponding Jacobian function and check that your parametrization is regular.

Question b

Now calculate the tangential line integral

\[\begin{equation*} \int_\mathcal{C}\pmb{V}\cdot \mathrm{d} \pmb{s}. \end{equation*}\]

4: Line Integral of Vector Field II. By Hand

In \((x,y,z)\) space we are given a vector field

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

as well as a curve \(\mathcal{C}\) with the parametric representation

\[\begin{equation*} \pmb{r}(u)=(u,u^2,u^3), \quad u\in\left[ 0,1\right] . \end{equation*}\]

Question a

Argue that \(\pmb{r}\) is a regular \(C^1\) parametrization.

Question b

Calculate the tangential line integral

\[\begin{equation*} \int_\mathcal{C}\pmb{V}\cdot \mathrm{d} \pmb{s}. \end{equation*}\]

5: Integration of Vector Field along Stair Line

In the plane we consider an arbitrary point \(\pmb{x}=(x_1,x_2)\) as well as the vector field

\[\begin{equation*} \pmb{V}: \mathbb{R}^2 \to \mathbb{R}^2, \quad \pmb{V}(x_1,x_2)=(x_1x_2,x_1). \end{equation*}\]

Question a

Calculate the tangential line integral of \(\pmb{V}\) along the straight line \(\mathcal{C}\) from \(\pmb{x}_0=\pmb{0}\) to \(\pmb{x}\).

Answer

\[\begin{equation*} \int_{\mathcal{C}} \pmb{V} \cdot \mathrm{d} \pmb{s} = \frac{1}{3}x_1^2x_2+\frac{1}{2} x_2 x_1 \end{equation*}\]

Question b

By the stair line from \(\pmb{x}_0=\pmb{0}\) to \(\pmb{x}\) we mean the piece-wise straight line that passes from \((0,0)\) to the point \((x_1,0)\) and then from \((x_1,0)\) to \((x_1,x_2)\).

On a piece of paper with a \((x_1,x_2)\) coordinate system: Sketch the stair line for different choices of \(\pmb{x}\). Then, calculate the tangential line integral of \(\pmb{V}\) along the stair line \(\mathcal{T}\) from \(\pmb{x}_0=\pmb{0}\) to \(\pmb{x}\).

Hint

See more about the stair-line method in this weeks Python demoes.

Hint

\[\begin{equation*} \int_{\mathcal{T}} \pmb{V}\cdot \mathrm{d} \pmb{s} = \int_0^{x_1} V_1(u,0)\mathrm{d} u+\int_0^{x_2} V_2(x_1,u)\mathrm{d} u. \end{equation*}\]

Answer

\[\begin{equation*} \int_{\mathcal{T}} \pmb{V}\cdot \mathrm{d} \pmb{s} = x_1 x_2. \end{equation*}\]

Question c

Determine, based on your answer to questions a and b, whether \(\pmb{V}\) is a gradient vector field.

Answer

The answer is no since the line integral is not path-independent.

Question d

There is an easier way to determine whether a vector field is a gradient vector field (at least when the vector field is defined on all of \(\mathbb{R}\)). What is this method?

Hint

Determine the Jacobian matrix of \(\pmb{V}\). Find the two lemmas in the textbook that are relevant.

Answer

For vector fields on an open and simply connected domain (as in this exercise) the vector field is a gradient vector field if and only if the Jacobian matrix is symmetric.

6: The Antiderivative Problem in \(\mathbb{R}^3\)

In 3D space we consider an arbitrary point \(\pmb{x}=(x_1,x_2,x_3)\), the vector field

\[\begin{equation*} \pmb{V}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \pmb{V}(x_1,x_2,x_3)=\begin{bmatrix} x_2\cos (x_1 x_2) \\ x_3+x_1 \cos (x_1x_2) \\ x_2 \end{bmatrix} \end{equation*}\]

and the vector field

\[\begin{equation*} \pmb{W}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \pmb{W}(x_1,x_2,x_3)= \frac{1}{1+x_1^2x_2^2+2x_1 x_2x_3^2+x_3^4} \begin{bmatrix} x_2 \\ x_1 \\ 2x_3 \end{bmatrix}. \end{equation*}\]

Question a

Determine the Jacobian matrix of \(\pmb{V}\). Is \(\pmb{V}\) a gradient vector field?

Question b

State all antiderivatives of \(\pmb{V}\).

Hint

Calculate the tangential line integral of \(\pmb{V}\), for instance along the stair line from \(\pmb{0}\) to \(\pmb{x}\).

Hint

By the stair line from \(\pmb{0}\) to \(\pmb{x}\) we mean the piece-wise straight line that passes from \((0,0,0)\) to the point \((x_1,0,0),\) then from \((x_1,0,0)\) to \((x_1,x_2,0)\) and finally from \((x_1,x_2,0)\) to \((x_1,x_2,x_3)\).

Hint

See more about the stair-line method in this week’s Python demoes.

Answer

The line integral is

\[\begin{equation*} \int_{\mathcal{T}} \pmb{V}\cdot \mathrm{d} \pmb{s} = \sin(x_1x_2)+x_2x_3, \end{equation*}\]

and all antiderivatives are thus

\[\begin{equation*} f(x_1,x_2,x_3)=\sin(x_1x_2)+x_2x_3+C, \quad C \in\mathbb{R}. \end{equation*}\]

Question c

Calculate using Sympy the tangential line integral of \(\pmb{W}\) along a straight line from \(\pmb{0}\) to the arbitrary point \(\pmb{x}\).

Question d

Investigate whether \(\pmb{W}\) is a gradient vector field, and if so then state all antiderivatives.

Answer

Yes, \(\pmb W\) is a gradient vector field. This can be seen by simply testing if \(\nabla f = \pmb{V}\). All antiderivatives are

\[\begin{equation*} f(x_1,x_2,x_3)=\arctan(x_1 x_2+x_3^2)+C,\quad C \in\mathbb{R}. \end{equation*}\]

7: Vector Field over a Circular Disc

Let \(U = \{ (x,y) \mid \frac{1}{4} < x^2 + y^2 < 1 \}\) be given. Consider the vector field

\[\begin{equation*} \pmb{V}: U \to \mathbb{R}^2, \quad \pmb{V}(x,y)= \frac{1}{x^2+y^2} \begin{bmatrix} -y \\ x \end{bmatrix}. \end{equation*}\]

Question a

Is the domain \(U\)…

1. open?

2. bounded?

3. curve connected?

4. simply connected?

5. star-shaped?

Answer

1. Yes

2. Yes

3. Yes

4. No

5. No

Question b

Determine whether \(\pmb{V}\) is \(C^0\) and \(C^1\). Find the Jacobian matrix for \(\pmb{V}\) and determine whether it is symmetric.

Answer

\(\pmb{V}\) is both \(C^0\) and \(C^1\). In fact, the vector field is \(C^\infty\) on \(U\). The Jacobian matrix is symmetric, since both non-diagonal terms can be written as \(-(x - y)(x + y)/(x^2 + y^2)^2\) (it can be helpful to use factor() on the terms).

Question c

Find the gradient of the arcustangent function \(f(x,y) = \mathrm{atan2}(y,x)\). The function is in SymPy given by f = atan2(y,x) and is a variant of \(\arctan(y/x)\).

Question d

Plot the function \(f\) on \(U\). Is \(f(x,y)\) an antiderivative of \(\pmb{V}\)?

Answer

No. Even though \(\pmb{V}\)’s Jacobian matrix is symmetric, the domain is not star-shaped which can cause trouble for the existence of antiderivatives, even for vector fields with symmetric Jacobian matrices. The problem with the chosen function \(f\) is that we do not have a reasonable definition along \(x=0\) due to the sign-switch on \(y\). Antiderivatives are differentiable and thus continuous, but the function \(f\) is not continuous (and thus not differentiable) for \(x=0\), where \(y\) intersects the \(x\) axis.

8: A Very Long Curve

The linear spiral curve \(\mathcal{C}\) in \(\mathbb{R}^2\) is parametrized by \(\pmb{r}: [0,1] \to \mathbb{R}^2\) where

\[\begin{equation*} \pmb{r}(u) = \begin{cases} (0,0) & \text{for } u = 0 \\ (u \cos(1/u), u \sin(1/u)) & \text{for } u \in \,]0,1]. \end{cases} \end{equation*}\]

Note that the domain of \(\pmb{r}\) is bounded and closed (just as we are used to). It can furthermore be shown that the curve is continuous, so \(C^0\), but not \(C^1\). In the textbook, chapter 7, it is required that the curves are \(C^1\), as unexpected things can happen when the curves are not \(C^1\). We want to illustrate that in this exercise.

Question a

Plot the curve. Show (you may use Python) that the norm of the tangent vector is

\[\begin{equation*} \Vert \pmb{r}'(u) \Vert = \sqrt{1 + u^{-2}} \end{equation*}\]

for \(u \,\,\in ]0,1]\).

Question b

Let \(\epsilon < 1\). Calculate the length \(\ell_\epsilon\) of the curve \(\pmb{r}(u)\) for \(u \in [\epsilon,1]\). Find \(\lim_{\epsilon \to 0} \ell_\epsilon\). What is the length \(\ell_0\) of the curve \(\mathcal{C}\)?

Exercises – Short Day

1: Surface Area of a Sphere

We consider a hollow sphere (a spherical shell) in \(\mathbb{R}^3\) centered at \((0,0,0)\) with a radius of \(a > 0\):

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

The surface area of the sphere is (as is known from elementary school) \(4 \pi a^2\). Find this expression again using a surface integral and a parametrization of the sphere.

2: Flux through Parametric Surfaces. By Hand

We are given the vector field

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

as well as the surface \(\mathcal{F}\) 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 corresponds to the parametric representation. Argue that the parametric representation is regular. Then, calculate the flux of the vector field through the surface.

Answer

The flux of \(\pmb{V}\) through \(\mathcal{F}\) is:

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

Question b

What meaning does the sign of the flux have? Can you switch the sign of the flux by changing the surface parametrization?

Question c

We are given the vector field

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

as well as the surface \(\mathcal{F}\) 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 parametric representation. Argue that the parametric representation 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 the so-called Coulomb vector field is known:

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

Note that the Coulomb vector field cannot be defined on all of \(\mathbb{R}^3\). We can, though, define it on the domain \(U = \mathbb{R}^3\setminus \{(0,0,0)\}\), which is open, and this is the default assumption made for vector fields in the textbook.

A massive cylinder \(B\) with a height of \(2h\) and a diameter of \(2a\), 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 \(B\) (easiest with pen and paper) and provide a parametric representation of each of the three parts that the boundary \(\partial B\) af \(B\) consists of: the bottom surface, the top surface, and the cylindrical wall.

Hint

Since we are dealing with surfaces, your parametric representations ought to use two parameters.

Question b

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

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

by calculating the flux through each of the three pieces that \(\partial B\) consists of and then adding those three result together. How does the size of the cylinder influence the flux? And what is the limit of the flux for \(a\) and \(h\) going towards \(0\)?

Hint

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

4: Flux via Gauss’ Divergence Theorem. Optional

Gauss’ divergence theorem is not part of the curriculum, but in this exercise we will still get acquainted with it as it is a key result in the study of vector fields that may be useful in some project topics:

Theorem (Gauss’ Divergence Theorem): 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 within a fluid, then the divergence measures the infinitesimal expansion- or contraction-rate of the fluid. The velocity field of an incompressible fluid has a divergence of zero.

Given the \(C^1\) vector field

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

as well as 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 boundary surface \(\,\partial \Omega\,\) is given an orientation defined by 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: half of a spherical shell and a circular base surface.

Answer

If Gauss is right, then you should be getting the same result in both questions. The answer is:

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

Question c

For which \(\,a\,\) is the flux with the given normal vector positive (meaning, “the flow out through \(\partial \Omega\) is greater than the flow in through it”).

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 vector field’s surface integral be considered 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 of 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 consists of the endpoints \(a\) and \(b\), whereas it for a volume integral over a region is the boundary of that region.

5: Flow Curves of a Vector Field. Optional

A linear vector field \(\pmb V\) i the \((x,y)\) plane is given by

\[\begin{equation*} \pmb V(x,y)=\left(\frac 18x +\frac 38y,\frac 38x +\frac 18y\right). \end{equation*}\]

We imagine that we at time \(t=0\) throw a particle into a force field at the point \((x_0,y_0)\), and we want to model the trajectory (curve) \(\pmb{r}(t)\) of the particle as a function of time. Such curves are known as integral curves or flow curves. They are solutions to the differential equation system:

\[\begin{equation*} \pmb{r}'(t) = \pmb V(\pmb{r}(t)), \quad \pmb{r}(0) = \pmb{x}_0, \end{equation*}\]

where \(\pmb{x}_0\) is the initial point.

This week’s Python demoes can be of help in this exercise.

Question a

Determine the matrix of the differential equation system and find (you may use Sympy’s eigenvects()) the eigenvalues and corresponding eigenvectors of the matrix.

Hint

The vector field is written in matrix form \(\displaystyle{\pmb{V}= A \begin{bmatrix}x\\ y\end{bmatrix}}\) where \(A\) is the system’s matrix.

Question b

The integral curve \(\pmb{r}_1(u)\) is determined by passing through the point \((0,-1)\) at time \(u=0,\) and the integral curve \(\pmb{r}_2(u)\) by passing through \((0,\frac 12)\) at time \(u=0\). Find, using the results from question a and the given initial conditions, a parametric representation for \(\pmb{r}_1(u)\) and \(\pmb{r}_2(u)\).

Answer

\(\displaystyle{\pmb{r}_1(u)=\begin{bmatrix}1/2\exp(-1/4u)-1/2\exp(1/2u)\newline -1/2\exp(-1/4u)-1/2\exp(1/2u)\end{bmatrix}}\)

\(\displaystyle{\pmb{r}_2(u)=\begin{bmatrix}1/4\exp(1/2u)-1/4\exp(-1/4u)\newline 1/4\exp(1/2u)+1/4\exp(-1/4u)\end{bmatrix}}\)

Question c

Make a Python illustration that contains the vector field and the two integral curves.