\\\\( \nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\\\)

Exercise 1: Determination of the Indefinite Integral. By Hand

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

$$\mV(x,y,z)=(x+z,-y-z,x-y)\,.$$
A

Compute $\,\mathrm{Curl}(\mathbf V)(x,y,z)\,$ and justify that $\,\mV\,$ is a gradient vector field.

B

Determine, using the tangential line integral of $\,\mV\,$ along the stair line to an arbitrary point $\,(x,y,z)\,$, the indefinite integral of $\,\mV\,.$

Exercise 2: Divergence and Curl. By Hand

A

Compute both the divergence and the curl of the vector field

$$\mathbf V(x,y,z)=(-y\,x\,,x\,y^2\,,x\,y\,z)$$

at the point $(1,1,1)$.

Exercise 3: Explosion, Rotation and Implosion Fields

In $(x,y,z)$ space we are given the vector fields

$$\begin{align*} \mathbf V(x,y,z)&=(-y,x,1)\,\newline \mathbf W(x,y,z)&=(-x,-y,-z)\,\newline \mathbf U(x,y,z)&=(x,y,z) \,. \end{align*}$$
A

We are informed that among these vector fields one is an explosion field, one is an implosion field, and one is a rotation field.

From just looking at the expressions, guess which is which. Discuss with a fellow student which properties and behaviour you would expect each to have. Try to guess as much information as you can, such as signs and values, about their divergences and curls before computing those properties in Question c) below.

B

Predict how the plots of the vector fields will look. Then plot each vector field in Maple.

C

Compute the divergence and curl of all three vector fields.

D

Which of the vector fields $\,\mU\,,\,\mV\,$ og $\,\mW\,$ are gradient vector fields?

Exercise 4: Tangential Line Integrals of Gradient Vector Fields

We are given a function

$$\,f(x,y,z)=\cos (x\,y\,z)\,,$$

a vector field

$$\mV(x,y,z)=\nabla f(x,y,z),$$

and a curve $\,\mathcal K\,$ that is the straight line from $\,\left(\pi,\frac{1}{2},0\right)\,$ to $\,\left(\frac{1}{2},\pi,-1\right)\,.$

A

Compute the tangential line integral

$$\int_{\mathcal K}\mV\cdot\me\, \mathrm d\mu.$$

We are now given the vector field

$$\mV(x,y,z)=\nabla (x^2+yz)$$

and a curve $\,\mathcal K\,$ with the parametric representation

$$\mr(u)=(\cos(u) ,\sin(u),\sin (2u)), \quad u\in\left[\, 0,2\pi\right] \,.$$
B

Compute the tangential line integral

$$\int_{\mathcal K}\mV\cdot\me\, \mathrm d\mu.$$

Exercise 5: Circulations in the Plane

In the $\,(x,y)$ plane we consider the circles

$$C_1:\,x^2+y^2=1 \\\\ C_2:\,(x-1)^2+(y-1)^2=1\,.$$

Moreover the vector field

$$\,\mV(x,y)=(x^2+y^2,xy)\,$$

is given.

A

Compute the tangential line integral of $\mV$ along one full trip counterclockwise along the two given circular paths.

B

Compute the tangential line integral of $\mV$ again, but this time clockwise one full trip around the two circles.

C

Can you from your results from the questions above determine whether $\,\mV$ is a gradient vector field?

Opg 6: Study of the Divergence (Advanced)

In this exercise we will – with a simple example of a vector field of the first degree with constant divergence – verify the following statement: “The divergence indicates the local “tendency of expansion” of the vector space.”

Follow the steps in the questions below. Along the way you will parametrize a spatial region in motion and determine its time-dependent volume!

In $\,(x,y,z)$ space the vector field

$$\,\mathbf V(x,y,z)=(5x-4z\,,-2x-y+4z\,,\,2x-z)\,$$

is given together with the axes-parallel cube

$$\mathcal A=\left\{(x,y,z)\,|\,-\frac 12\,\leq x \leq \frac 12\,\,,1\leq\,y\leq 2\,\,,-\frac 12\,\leq z \leq \frac 12\,\right\}\,.$$
A

Determine (readily using dsolve) the flow curve $\,\mr(t)\,$ of $\,\mathbf V\,$ that corresponds to the initial condition that $\,\mr(0)\,$ is an arbitrary point within $\,\mathcal A\,.$

Let $\,\mathcal A_t\,$ be the solid that $\,\mathcal A\,$ is deformed into at time $\,t\,$ if we imagine that $\,\mathcal A\,$ flows with the vector field.

B

Provide an expression of the volume $\mathrm{Vol}(t)\,$ of $\,\mathcal A_t\,$ expressed as a function of $\,t\,.$

C

How large is the volume at time $\,t=1\,?$

D

Determine the ratio

$$\frac{\mathrm{Vol}\,'(0)}{\mathrm{Vol}(0)}\,\,,$$

and compute $\mathrm{Div}(\mathbf V)(x,y,z)\,$.