Exercise 1: Determination of an Indefinite Integral. By Hand
In the $\,(x,y,z)$-space is given the vector field
$$\mV(x,y,z)=(x+z,-y-z,x-y)\,.$$
A
Determine $\,\mathbf{Curl}(\mathbf V(x,y,z)\,)\,$ and argue that $\,\mV\,$ is a gradient vector field.
hint
See Theorem 27.14 in eNote 27.
B
Determine using the tangential curve integral of $\,\mV\,$ along the stair line to an arbitrary point $\,(x,y,z)\,$ all indefinite integrals $\,\mV\,.$
Div$(\mathbf V(x,y,z)\,)=2$$ $ and $ $$\mathbf{Curl}(\mathbf V(x,y,z)\,)=(1,-1,2)\,.$
Exercise 3: Explosion, Implosion and Rotation
In the $(x,y,z)$-space are given the vector fields
\begin{align}
\mathbf U(x,y,z)&=(x,y,z)\,,\
\mathbf V(x,y,z)&=(-y,x,1)\,,\
\mathbf W(x,y,z)&=(-x,-y,-z)\,.
\end{align}
A
Determine the divergence and curl of the three vector fields.
answer
Div$(\mathbf U(x,y,z)\,)=3$$ $ and $ $$\mathbf{Curl}(\mathbf U(x,y,z)\,)=(0,0,0)$
Div$(\mathbf V(x,y,z)\,)=0$$ $ and $ $$\mathbf{Curl}(\mathbf V(x,y,z)\,)=(0,0,2)$
Div$(\mathbf W(x,y,z)\,)=-3$$ $ and $ $$\mathbf{Curl}(\mathbf W(x,y,z)\,)=(0,0,0)$
B
Decide on the basis of your answer to Question A) which of the vector fields $\,\mU\,,\,\mV\,$ og $\,\mW\,$ that are gradient vector fields.
hint
See Theorem 27.14 in eNote 27 (the curl of a gradient field is zero!)
answer
$\,\mU\,$ and $\,\mW\,$ are gradient vector fields, while $\,\mV\,$ is not.
Exercise 4: Tangential Curve Integrals in Gradient Vector Fields
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 $\,K\,$ that is the straight line from $\,(\pi,\frac{1}{2},0)\,$ to $\,(\frac{1}{2},\pi,-1)\,.$
A
Determine the tangential curve integral
$$\int_K\mV\cdot\me\, d\mu.$$
hint
Is it necessary to compute the integral?
hint
It is only necessary to compute a couple of functional values of $\,f\,$. See Theorem 27.10 in eNote 27.
Determine the tangential curve integral of $\mV$ along the circles when the circles are run through one time counter-clockwise.
hint
You shall need a parametric representation for each of the circles.
hint
For $C_1$ you can use $\mr(u)=(\cos(u),\sin(u))\,,\,\,u\in \left[0,2\pi\right]\,.$
For $C_2$ you can use $\mr(u)=(\cos(u)+1,\sin(u)+1)\,,\,\,u\in \left[0,2\pi\right]\,.$
Determine the tangential curve integral of $\mV$ along the circles when the circles are run through one time clockwise.
hint
You shall need a parametric representation for each of the circles.
hint
For $C_1$ you can use $\mr(u)=(\cos(u),-\sin(u))\,,\,\,u\in \left[0,2\pi\right]\,.$
For $C_2$ you can use $\mr(u)=(\cos(u)+1,-\sin(u)+1)\,,\,\,u\in \left[0,2\pi\right]\,.$
From the results of Questions a) and b) : Is $\,\mV$ a gradient vector field?
answer
Here you must be careful: The integral along a closed curve of a gradient vector field is 0, but you may readily find that an integral along a closed curve of a vector field is 0, even though the vector field is not a gradient vector field. Here $\,\mV\,$ is not a gradient vector field, since the integral along $\,C_2\,$ is different from 0.
Opg 6: Study of the Divergence (advanced)
In this exercise we will – in a simple example (with a vector field of first degree and constant divergence) – verify the following statement: The divergence states the local ‘‘tendency of expansion’’ in the vector space. Just follow the steps below - along the way you shall parametrize a spatial field in motion and determine its time dependent volume!
$$\mathcal A=\left\{(x,y,z)\,|\,-\frac 12\,\leq x \leq \frac 12\,\,,1\leq\,y\leq 2\,\,,-\frac 12\,\leq z \leq \frac 12\,\right\}\,.$$
A
Determine (readily using dsolve) the flow curve $\,\mr(t)\,$ for $\,\mathbf V\,$ corresponding to the initial condition that $\,\mr(0)\,$ is an arbitrary point in $\,\mathcal A\,.$
Let $\,\mathcal A_t\,$ the solid that $\,\mathcal A\,$ is deformed into at time $\,t\,$ when we imagine that $\,\mathcal A\,$ flows with the vector field.
B
Find an expression for the volume Vol$(t)\,$ of $\,\mathcal A_t\,$ expressed as a function of $\,t\,.$ How large is the volume at time $\,t=1\,.$ Determine the ratio
