Section6.5Constructing Functions
¶We now know how to draw a vector field provided someone tells us the equation. How do we obtain an equation of a vector field? The following problem will help you develop the gravitational vector field.
Exercise6.5.1Radial fields
(a)
Let \(P=(x,y,z)\) be a point in space. At that point, what is the \(x\text{,}\) \(y\text{,}\) and \(z\) distance to the origin?
(b)
At the point \(P\text{,}\) let \(\vec F(x,y,z)\) be the vector which points from \(P\) to the origin. Give a formula for \(\vec F(x,y,z)\text{.}\)
HintAn object following this vector would travel from the point to the origin
(c)
Give an equation of the vector field where at each point \(P\) in the plane, the vector \(\vec F_2(P)\) is a unit vector that points towards the origin.
(d)
Give an equation of the vector field where at each point \(P\) in the plane, the vector \(\vec F_3(P)\) is a vector of length 7 that points towards the origin.
Use the SageMath Cell below to plot your vector fields. Note: The current input is NOT the same as the field above, modify it to reflect the fields you found.
If someone gives us parametric equations for a curve in the plane, we know how to draw the curve. How do we obtain parametric equations of a given curve? In Exercise 6.2.1, we were given the parametric equation for the path of a horse, namely \(x=2\cos t, y=3 \sin t\) or \(\vec r(t)=(2\cos t,3\sin t)\text{.}\) From those equations, we drew the path of the horse, and could have written a Cartesian equation for the path. How do we work this in reverse, namely if we had only been given the ellipse \(\ds\frac{x^2}{4}+\frac{y^2}{9}=1\text{,}\) could we have obtained parametric equations \(\vec r(t)=(x(t),y(t))\) for the curve?
Exercise6.5.2
Give a parametrization of the straight line from \((a,0)\) to \((0,b)\text{.}\) You can write your parametrization in the vector form \(\vec r(t)=(?,?)\text{,}\) or in the parametric form \(x=?,\ y=?\text{.}\) Remember to include bounds for \(t\text{.}\)
Exercise6.5.3
Give a parametrization of the parabola \(y=x^2\) from \((-1,1)\) to \((2,4)\text{.}\) Remember the bounds for \(t\text{.}\)
Exercise6.5.4
Give a parametrization of the function \(y=f(x)\) for \(x\in[a,b]\text{.}\) You can write your parametrization in the vector form \(\vec r(t)=(?,?)\text{,}\) or in the parametric form \(x=?,\ y=?\text{.}\) Include bounds for \(t\text{.}\)
If someone gives us parametric equations for a surface, we can draw the surface. This is what we did in Exercise 6.3.1 and Exercise 6.3.5. How do we work backwards and obtain parametric equations for a given surface? This requires that we write an equation for \(x\text{,}\) \(y\text{,}\) and \(z\) in terms of two input variables (See Exercise 6.3.1 and Exercise 6.3.5 for examples). In vector form, we need a function \(\vec r\colon \mathbb{R}^2\to\mathbb{R}^3\text{.}\) We can often use a coordinate transformation \(\vec T\colon \mathbb{R}^3\to\mathbb{R}^3\) to obtain a parametrization of a surface.
The next three problems show how to do this.
Exercise6.5.5
(a)
Using the rectangular coordinate transformation \(\vec T(x,y,z)=(x,y,z)\text{,}\) give a parametrization \(\vec r\colon \mathbb{R}^2\to\mathbb{R}^3\) of the surface. This is the same as saying
\begin{equation*}
x=x, y=y, z=?.
\end{equation*}
HintUse the surface equation to eliminate the input variable \(z\) in \(T\text{.}\)
(b)
What bounds must you place on \(x\) and \(y\) to obtain the portion of the surface above the plane \(z=0\text{?}\)
(c)
If \(z=f(x,y)\) is any surface, give a parametrization of the surface (i.e., \(x=?, y=?, z=?\) or \(\vec r (?,?)=(?,?,?)\text{.}\))
Use Wolfram Alpha or the SageMath cell below to plot the parameterizations.
Exercise6.5.6
Again consider the surface \(z=9-x^2-y^2\text{.}\)
(a)
Using cylindrical coordinates, \(\vec T(r,\theta,z) = (r\cos \theta, r\sin\theta, z)\text{,}\) obtain a parametrization \(\vec r(r,\theta)=(?,?,?)\) of the surface using the input variables \(r\) and \(\theta\text{.}\) In other words, if we let \(x=r\cos \theta, y=r\sin\theta, z=z\text{,}\) write \(z=9-x^2-y^2\) in terms of \(r\) and \(\theta\text{.}\)
(b)
What bounds must you place on \(r\) and \(\theta\) to obtain the portion of the surface above the plane \(z=0\text{?}\)
Use the SageMath Cell or Wolfram Alpha to plot your parametrization with your bounds (See Exercise 5 for examples).
Exercise6.5.7
Recall the spherical coordinate transformation as given in Exercise 6.4.9.
\begin{equation*}
\vec T(\rho,\phi,\theta) = (\rho\sin\phi\cos \theta, \rho\sin\phi\sin \theta,\rho \cos \phi).
\end{equation*}
This is a function of the form \(\vec T\colon \mathbb{R}^3\to\mathbb{R}^3\text{.}\) If we hold one of the three inputs constant, then we have a function of the form \(\vec r\colon \mathbb{R}^2\to\mathbb{R}^3\text{,}\) which is a parametric surface.
(a)
Give a parametrization of the sphere of radius 2, using \(\phi\) and \(\theta\) as your input variables.
(b)
What bounds should you place on \(\phi\) and \(\theta\) if you want to hit each point on the sphere exactly once?
HintThere are two possible answers here
(c)
What bounds should you place on \(\phi\) and \(\theta\) if you only want the portion of the sphere above the plane \(z=1\text{?}\)
HintThere are also two possible answers here
Use the SageMath Cell or Wolfram Alpha to plot your parametrization with your bounds (See Exercise 5 for examples).
Sometimes you'll have to invent your own coordinate system when constructing parametric equations for a surface. If you notice that there are lots of circles parallel to one of the coordinate planes, try using a modified version of cylindrical coordinates. Instead of circles in the \(xy\) plane (\(x=r\cos\theta,y=r\sin\theta,z=z\)), maybe you need circles in the \(yz\)-plane (\(x=x,y=r\sin\theta,z=r\sin\theta\)) or the \(xz\) plane. Just look for lots of circles, and then construct your parametrization accordingly.
Exercise6.5.8
(a)
Find parametric equations for the surface \(x^2+z^2=9\text{.}\)
HintRead the paragraph above.
(b)
What bounds should you use to obtain the portion of the surface between \(y=-2\) and \(y=3\text{?}\)
(c)
What bounds should you use to obtain the portion of the surface above \(z=0\text{?}\)
(d)
What bounds should you use to obtain the portion of the surface with \(x\geq 0\) and \(y\in[2,5]\text{?}\)
Use the SageMath Cell or Wolfram Alpha to plot your parametrization with your bounds (See Exercise 5 for examples).
Exercise6.5.9
Construct a graph of the surface \(z = x^2-y^2\text{.}\) Do so in 2 ways. (1) Construct a 3D surface plot. (2) Construct a contour plot, which is a graph with several level curves. Which level curve passes through the point \((3,4)\text{?}\) Use Wolfram Alpha to know if you're right. Just type “plot z=x\^2-y\^2.”
Exercise6.5.10
Construct a plot of the vector field
\begin{equation*}
\vec F(x,y) = (x+y, -x+1)
\end{equation*}
by graphing the field at many integer points around the origin (I generally like to get the 8 integer points around the origin, and then a few more). Then explain how to modify your graph to obtain a plot of the vector field
\begin{equation*}
\hat F(x,y) = \frac{(x+y, -x+1)}{\sqrt{(x+y)^2+(1-x)^2}}.
\end{equation*}