Determine the system matrix that corresponds to $\mathbf U$ and find (readily using Maple’s Eigenvectors command) the eigenvalues of the matrix and the corresponding eigenvectors.
hint
The vector field is on matrix form written as $\,\displaystyle{\mU=\mA\,\begin{matr}{c}x\newline y\end{matr}}\,$ where $\,\mA\,$ is the system matrix.
The flow curve $\,\mr_1(u)\,$ is determined by passing through the point $\,(0,-1)\,$ at time $\,u=0\,,$ and the flow curve $\,\mr_2(u)\,$ by passing through $\,(0,\frac 12)\,$ at time $\,u=0\,$. State using these initial conditions and the result from question a) a parametric representation of $\,\mr_1(u)\,$ and $\,\mr_2(u)\,.$
Find in eNote 17 the relevant solution formula for the case of real and different eigenvalues of the system matrix of this linear differential equation system.
hint
Remember to use the given initial conditions to find the exact values of the constants that appear in the solution formula. It is only with these initial conditions that the two flow curves separate themselves from each other.
Compute the eigenvalues of the system matrix that corresponds to $\,\mathbf V\,$.
answer
$$\lambda=\frac 1{10},i,-i$$
B
Determine the flow curve $\,\mr(u)\,$ of $\,\mathbf V\,$ that corresponds to the initial condition $\,\mr(0)=(1,1,1)\,$ (if you feel comfortable about the solution formulas for differential equation systems, then use Maple’s dsolve command to solve this).
answer
$$\mr(u)=\begin{matr}{c}\sin(u)+\cos(u)\newline \e^{\frac{1}{10}\,u}\newline \cos(u)-\sin(u)\end{matr}\quad , u \in \Bbb R$$
C
Create 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
Provide a parametric representation of $\,\mathcal L\,.$
answer
A possibility is
$$(v,v,v)\quad, v \in [1,2]\,.$$
E
Determine (readily using dsolve) the flow curve $\,\mathbf s(u)\,$ of $\,\mathbf V\,$ corresponding to the initial condition that $\,\mathbf s(0)\,$ is an arbitrary point on $\,\mathcal L\,$.
hint
With the above parametric representation of $\mathcal L$, the initial condition is
$$\,\mathbf s(0)=(v,v,v)\,.$$
Use it to find a flow curve in the same way as usual, just with this varying initial condition.
F
Provide a parametric representation $\mathbf k(u,v)$ of the surface $\,\mathcal F\,$ that $\,\mathcal L\,$ sweeps through when we imagine that $\,\mathcal L\,$ flows with the vector field through the time $\,u\in [\,0,5\,\pi\,]\,.$ Plot $\,\mathcal F\,$ together with the vector field.
answer
With the above parametric representation of $\,\mathcal L\,$, the surface is parametrized as:
$$\mathbf k(u,v)=\begin{matr}{c}v\sin(u)+v\cos(u)\newline v\e^{\frac{1}{10}\,u}\newline v\cos(u)-v\sin(u)\end{matr}\quad, \, u \in [0,5 \pi],v\in [1,2]\,.$$
Exercise 3: Gradient Vector field in 3D Space
A function $f$ is given by
$$f(x,y,z)=(x-1)^2+2\,(y-1)^2+(z-1)^2-4\,.$$
A
Compute by hand (or in your head) the gradient of $\,f\,.$
answer
$$\nabla f(x,y,z)=(2x-2,4y-4,2z-2)$$
B
Find a parametric representation of the level surface$\,\mathcal K_0\,$ that corresponds 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. State its three semi axes by comparing this equation with the general form in the table in the Section 22.3 in eNote 22. With that it is easy to create a parametric representation - see examples in Section 22.4 in eNote 22. Remember that its centre is $\,(1,1,1)\,.$
The gradient of $\,f\,$ can be interpreted as a gradient vector field in $(x,y,z) $space. Create a Maple illustration that contains both a plot of the level surface $\,\mathcal K_0\,$ and a plot of the gradient vector field.
hint
Use the fieldplot3d command.
D
Show that the gradient of $\,f\,$ at every point $\,P\,$ on $\,\mathcal K_0\,$ is perpendicular to $\,\mathcal K_0\,$ at $\,P\,$ (or more precisely, perpendicular to the tangent plane to $\,\mathcal K_0\,$ at $\,P\,).$
hint
Check the dot product.
E
Discuss with a student what the meaning is of the well-known slogan: “The gradient points in the direction along which the function increases the most” in this context.
One of these vector fields is a gradient vector field, the other is not. Show that this statement is correct.
hint
There are a few different approaches to showing that a vector field is a gradient vector field. One such approach is to show that the vector field has an antiderivative. Try to guess an antiderivative; this might be fairly easy for one of the fields.
hint
For the other vector field, see 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 Line Integrals. By Hand
In the $(x,y)$ plane we are given a vector field
$$\mV(x,y)=(x^2-2xy\,,\,y^2-2xy)$$
and a curve $\,\mathcal K\,$ by the equation
$$y=x^2, \quad x\in\left[ -1,1\right]\,.$$
A
Compute the tangential line integral
$$\int_{\mathcal K}\mV\cdot\me\, \mathrm d\mu.$$
hint
First create a parametrization $\mr(u)$ of $\mathcal K$.
In the plane we consider an arbitrary point $\,P=(x,y)\,$ and
the vector field
$$\,\mV(x,y)=(xy,x)\,.$$
A
Compute the tangential line integral of $\,\mV\,$ along the straight line from the origin to $\,P\,.$
hint
The fact that the end point is an arbitrary point and thus expressed in terms of $x$ and $y$ doesn’t matter. Just use the same approach and formula as in the previous exercises.
hint
A parametrization of the straight line could be
$$(ux,uy),u \in [0,1]\,.$$
answer
$$\frac{1}{3}x^2y+\frac{1}{2}yx$$
By the stair line from the origin to point $\,P\,$ we understand the piece-wise straight lines that run from the origin to point $\,(x,0)\,$ and then from $\,(x,0)\,$ to $\,(x,y)\,.$
B
On a piece of paper with an $\,(x,y)$ coordinate system sketch a stair line to a few chosen points.
C
Compute the tangential line integral of $\,\mV\,$ along a stair line from the origin to the arbitrary point $\,P\,.$
hint
See an example of the stair-line method in today’s Maple demo.
hint
You must parametrize each segment of the stair line. Then compute the tangential line integral along each and add them up.
Decide based on your answers to questions a) and c) above whether $\,\mV\,$ is a gradient vector field.
answer
$\mV\,$ is not a gradient vector field because integration depends on the chosen route through the field (the tangential line integrals are different along different routes).
Exercise 7: Scalar potentials
A vector field $\mathbf V$ is a gradient vector field if a function $f$ exists so that $\nabla f=\mathbf V$. If so, then $f$ is called an antiderivative or a scalar potential of the vector field. A vector field is a gradient vector field if and only if it has a scalar potential.
We consider in 3D space an arbitrary point $\,P=(x,y,z)\,$ and the vector field
$$\,\mV(x,y,z)=(y\cos (xy),z+x\cos (xy),y)\,.$$
A
Determine the tangential line integral of $\,\mV\,$ along a stair line from the origin to $\,P\,.$
hint
Use the approach from the previous exercise.
hint
By the stair line from the origin to $\,P\,$ we understand the piece-wise straight line from the origin to point $\,(x,0,0)\,$ and then from $\,(x,0,0)\,$ to $\,(x,y,0)\,$ and finally from $\,(x,y,0)\,$ to $\,(x,y,z)\,.$ You must carry out the tangential line integral along all three pieces and sum up the results.
answer
$$\sin(xy)+yz$$
B
Investigate whether $\,\mV\,$ is a gradient vector field. If so, then state a scalar potential and state the indefinite integral.
hint
Check whether the gradient of the function you found in the previous question is identical to the vector field.
answer
$\mV\,$ is a gradient vector field. A scalar potential (an antiderivative) is
$$F(x,y,z)=\sin(xy)+yz.$$
The indefinite integral (all antiderivatives or all scalar potentials) is
