Determine the system matrix that corresponds to $\mathbf U$ and find (readily using Maple’s Eigenvectors) the eigenvalues of the matrix and the corresponding eigenvectors.
hint
The vector field is written in matrix form $\,\displaystyle{\mU=\mA\,\begin{matr}{c}x\\y\end{matr}}\,$ where $\,\mA\,$ is the system matrix.
B
The flow curve $\,\mr_1(u)\,$ is determined by going through the point $\,(0,-1)\,$ at the time $\,u=0\,,$ and the flow curve $\,\mr_2(u)\,$ by going through $\,(0,\frac 12)\,$ at the time $\,u=0\,$. Find using the results from Question a) and the given initial conditions a parametric representation for $\,\mr_1(u)\,$ og $\,\mr_2(u)\,.$
Make a Maple-illustration where $\,\mathbf V\,$ is plotted together with $\,\mr(u)\,$ for $\,u\in [\,0,5\,\pi\,]\,.$
Consider the line segment $\,\mathcal L\,$ extending from the point $\,(1,1,1)\,$ to the point $\,(2,2,2)\,.$
D
Determine a parametric representation for $\,\mathcal L\,.$
E
Determine (readily using dsolve) the flow curve $\,\mr(u)\,$ for $\,\mathbf V\,$ corresponding to the inial condition, that $\,\mr(0)\,$ is an arbitrary point on $\,\mathcal L\,$.
F
Determine a parametric representation for the surface $\,\mathcal F\,$ that $\,\mathcal L\,$ runs through in the space, when we imagine that $\,\mathcal L\,$ flows with the vector field in the time $\,u\in [\,0,5\,\pi\,]\,,$ and plot $\,\mathcal F\,$ together with the vector field.
answer
If the parametric representation for $\,\mathcal L\,$ is chosen as $\,(v,v,v)\,$ for $\,v\in \left[1,2\right]\,,$ the following can be used with $\,u\,$ and $\,v\,$ in the given intervals:
Show by hand (mental calculation ?) the gradient of $\,f\,.$
B
Find a parametric representation for the level surface$\,\mathcal K_0\,$ corresponding to $\,f(x,y,z)=0\,.$
hint
The equation for $\,\mathcal K_0\,$ is $\,(x-1)^2+2\,(y-1)^2+(z-1)^2=4\,.$ It is an ellipsoid, find its three semi-axes. Then it is easy to state its parametric representation, see e.g. Section 22.4 in eNote 22. Remember its centre is $\,(1,1,1)\,.$
The gradient of $\,f\,$ can be interpreted as a gradient vector field in space. Make a Maple-illustration that contains both a plot of the level surface $\,\mathcal K_0\,$ and a plot of the gradient vector field (use fieldplot3d).
D
Show that the gradient for $\,f\,$ in every point $\,P\,$ on $\,\mathcal K_0\,$ is perpendicular to $\,\mathcal K_0\,$ in $\,P\,$ (or more precisely: to the tangent plane to $\,\mathcal K_0\,$ in $\,P\,).$
E
What is the meaning of the well-known slogan: ‘‘The gradient points in the direction in which the function increases the most’’ in this context?
Exercise 4: Ponderings about Gradients
Two vector fields in space are given by:
\begin{align}
\mathbf U(x,y,z)&=(xy\cos(z)\,,y^2+xz\,,3z)\,\
\mathbf V(x,y,z)&=(2x\mathrm{e}^{x^2},2\cos(y^2)\,y\,,3)\,
\end{align}
A
The claim is that one of these vector fields is a gradient vector field while the other is not. Show that this statement is correct!
hint
For one of the vector fields you can possibly see/guess which function $\,f(x,y,z)\,$ for which the vector field is the gradient? For the other vector field, see a corresponding example from the plane: Example 26.3 in eNote 26.
answer
$\mathbf U(x,y,z)$ is not a gradient field, but $\mathbf V(x,y,z)$ is a gradient field.
Exercise 5: Tangential Curve Integrals. By Hand
In the $(x,y)$-plane is given a vector field
$$\mV(x,y)=(x^2-2xy\,,\,y^2-2xy)$$
and a curve $\,K\,$ by the equation
$$y=x^2, \quad x\in\left[ -1,1\right]\,.$$
A
Determine the tangential curve integral
$$\int_K\mV\cdot\me\, d\mu.$$
In the $\,(x,y,z)$-space is given a vector field
$$\mV(x,y,z)=(\,y^2-z^2\,,\,2yz\,,-x^2)$$
and a curve $\,K\,$ with the parametric representation
In the plane we consider an arbitrary point $\,P=(x,y)\,$ and
the vector field
$$\,\mV(x,y)=(xy,x)\,.$$
C
Determine the tangential curve integral of $\,\mV\,$ along the straight line from the origin to $\,P\,.$
answer
$\displaystyle{\int_K \mV\cdot \mathbf e \mathrm d\mu
=\frac{1}{3}x^2y+\frac{1}{2}yx}$
By the stair line from the origin to the point $\,P\,$ we understand the piecewise straight line that goes from the origin to the point $\,(x,0)\,$ and then from $\,(x,0)\,$ to $\,(x,y)\,.$
D
On a piece of paper with the $\,(x,y)$-coordinate system: Sketch the stair line for different choices of $\,P\,.$ Then determine the tangential curve integral of $\,\mV\,$ along the stair line from the origin to an arbitrary $\,P\,.$
hint
About the stair method: See the 3d-version in the MapleDemo.
Decide on the basis of you answers to Questions a) and b) whether $\,\mV\,$ is a gradient vector field.
answer
The answer is no, because $\ldots$
Exercise 7: Indefinite Integrals
We consider in space point $\,P=(x,y,z)\,$ and the vector field
$$\,\mV(x,y,z)=(y\cos (xy),z+x\cos (xy),y)\,.$$
By the stair line from the origin to $\,P\,$ we understand the piecewise straight line from the origin to the point $\,(x,0,0)\,,$ and then from $\,(x,0,0)\,$ to $\,(x,y,0)\,$ and finally from $\,(x,y,0)\,$ to $\,(x,y,z)\,.$
A
Determine the tangential curve integral of $\,\mV\,$ along the stair line from the origin to $\,P\,.$
