22. Parametric Surfaces and Surface Integrals

a. Introduction to Parametric Surfaces

1. Position and Plots

In a plane, \(\mathbb{R}^2\), the geometrical domains on which we can integrate are curves and regions.

A parametric curve is defined by an \(\mathbb{R}^2\) vector valued function of \(1\) variable: \[ \vec r(t)=\left\langle x(t),y(t)\right\rangle \]
A parametrized 2D region or 2D curvilinear coordinate system is defined by an \(\mathbb{R}^2\) vector valued function of \(2\) variables: \[ \vec R(u,v)=\left\langle x(u,v),y(u,v)\right\rangle \]

In space, \(\mathbb{R}^3\), the geometrical domains on which we can integrate are curves, surfaces and regions.

A parametric curve is defined by an \(\mathbb{R}^3\) vector valued function of \(1\) variable: \[ \vec r(t)=\left\langle x(t),y(t),z(t)\right\rangle \]
A parametric surface is defined by an \(\mathbb{R}^3\) vector valued function of \(2\) variables: \[ \vec R(u,v)=\left\langle x(u,v),y(u,v),z(u,v)\right\rangle \]
A parametrized 3D region or 3D curvilinear coordinate system is defined by an \(\mathbb{R}^3\) vector valued function of \(3\) variables: \[ \vec R(u,v,w)=\left\langle x(u,v,w),y(u,v,w),z(u,v,w)\right\rangle \]

We have studied how to integrate on all of these geometric domains except surfaces. That is the topic of this chapter. In some ways parametric surfaces behave like parametric curves and in some ways like curvilinear coordinate systems.

Position

A parametric curve gives the position on the curve as a function of \(1\) parameter, usually regarded as the time. \[ (x,y,z)=\vec r(t)=\left\langle x(t),y(t),z(t)\right\rangle \]

The parametrization can also be specified in component form: \[ x=x(t), \qquad y=y(t), \qquad z=z(t) \]

A curvilinear coordinate system gives the position in space as a function of \(3\) parameters, the coordinates. \[ (x,y,z)=\vec R(u,v,w)=\left\langle x(u,v,w),y(u,v,w),z(u,v,w)\right\rangle \]

The parametrization can also be specified in component form: \[ x=x(u,v,w), \qquad y=y(u,v,w), \qquad z=z(u,v,w) \]

A parametric surface gives the position on the surface as a function of \(2\) parameters, usually regarded as coordinates on the surface. \[ (x,y,z)=\vec R(u,v)=\left\langle x(u,v),y(u,v),z(u,v)\right\rangle \]

The parametrization can also be specified in component form: \[ x=x(u,v), \qquad y=y(u,v), \qquad z(u,v) \]

As a primary example of a parametric surface, we will study the sphere (centered at the origin) of radius \(2\), which may be parametrized as \[ (x,y,z)=\vec R(\phi,\theta) =\left\langle 2\sin\phi\cos\theta,2\sin\phi\sin\theta,2\cos\phi\right\rangle \] Notice that this is exactly the same as the spherical coordinate system, except that the radius \(\rho\) (the distance from the origin) has been replaced by \(2\). The coordinates (parameters) on the surface are
(1) the polar angle, \(\phi\), measured down from the \(+z\) axis (the North pole) to the \(-z\) axis (the South pole), and
(2) the azimuthal angle, \(\theta\), measured counterclockwise (West to East) from the \(+x\) axis.
Add xyz axes and \(\phi\) and \(\theta\) arrows to the plot.

Plot

The plot of a parametric curve is the image of the function \(\vec r(t)\), i.e. the set of points \(\left\langle x(t),y(t),z(t)\right\rangle\) for all values of \(t\). For example, the plot of the helix, \(\vec r(t)=\left\langle 4\cos t,4\sin t,3t\right\rangle\) is shown at the right.

Similarly, the plot of a parametric surface is the image of the function \(\vec R(u,v)\), i.e. the set of points \(\left\langle x(u,v),y(u,v),z(u,v)\right\rangle\) for all values of \(u\) and \(v\). For example, the plot of the sphere, \[ \vec R(\phi,\theta) =\left\langle 2\sin\phi\cos\theta,2\sin\phi\sin\theta,2\cos\phi\right\rangle \] is shown at the right.

Parameterizing a Surface

A surface is frequently given by an equation, either as the graph of a function of \(2\) variables, \(z=f(x,y)\), or as the level set of a function of \(3\) variables, \(F(x,y,z)=C\). At times, you may want (or need) to rewrite it in parametric form, for instance to be able to integrate on the surface. Frequently a good method is to pick an appropriate coordinate system (rectangular, cylindrical, spherical, etc), start with the position vector \(\vec R\) of that coordinate system, and then eliminate one of the coordinates by using the equation of the surface. The resulting formula for \(\vec R\) will have only two coordinates which means that you have a parametric surface. The choice of coordinate system will depend on the shape of the surface and the shape of the boundary of the piece of the surface you are interested in. The graph of a function, \(z=f(x,y)\) or \(z=f(r,\theta)\), can always be parametrized in rectangular or cylindrical coordinates by replacing \(z\) by \(f(x,y)\) or \(f(r,\theta)\); the choice between them will depend on the domain of the function and the shape of the boundary.

Rectangular coordinates are useful when the surface is given as the graph of a function over a region in the \(xy\)-plane given in rectangular coordinates.

Parametrize the piece of the elliptic paraboloid \(z=x^2+y^2\) above the rectangle \([0,2]\times[0,3]\). Give the range of the parameters (which will be the limits of integration in future problems).

If we let \(x=u\) and \(y=v\), then the equation of the surface is \(z=u^2+v^2\) and the parametrization is: \[ \vec R(u,v)=\left\langle u,v,u^2+v^2\right\rangle \] On the rectangle, \(0 \le x \le 2\) and \(0 \le y \le 3\). So the parameter ranges are: \[ 0 \le u \le 2 \qquad \text{and} \qquad 0 \le v \le 3 \] Actually, there was no reason to introduce the extra variables \(u\) and \(v\). We could simply give the parametrization as \[ \vec R(x,y)=\left\langle x,y,x^2+y^2\right\rangle \] with parameter ranges \(0 \le x \le 2\) and \(0 \le y \le 3\). In this form the parametrization is simply the rectangular coordinates \((x,y,z)\) with \(z\) replaced by \(x^2+y^2\).

Cylindrical coordinates are useful when the surface is a piece of a cylinder or or has axial symmetry or is given as the graph of a function given in cylindrical coordinates.

Parametrize the piece of the elliptic paraboloid \(z=x^2+y^2\) below \(z=9\). Give the range of the parameters.
PY: Change the grid on the parabola to polar.

Since the surface (\(z=x^2+y^2\)) and the boundary (\(x^2+y^2=9\)) are axisymmetric, it is useful to start with cylindrical coordinates: \[ \vec R(r,\theta,z)=\left\langle r\cos\theta,r\sin\theta,z\right\rangle \] In cylindrical coordinates, the equation of the surface is \(z=r^2\). So the parametrization of the surface becomes \[ \vec R(r,\theta)=\left\langle r\cos\theta,r\sin\theta,r^2\right\rangle \] Since the boundary is \(r^2=9\) and there is no restriction on \(\theta\), the parameter ranges are: \[ 0 \le r \le 3 \qquad \text{and} \qquad 0 \le \theta \le 2\pi \]

Spherical coordinates are useful when the surface is a piece of a sphere or has spherical symmetry or is given in spherical coordinates.

Parametrize the apple given in spherical coordinates by \(\rho=\phi\). Give the range of the parameters.

We start with spherical coordinates: \[ \vec R(\rho,\phi,\theta) =\left\langle \rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi\right\rangle \] Since \(\rho=\phi\), we replace \(\rho\) by \(\phi\). So the parametrization of the surface is: \[ \vec R(\phi,\theta) =\left\langle \phi\sin\phi\cos\theta,\phi\sin\phi\sin\theta,\phi\cos\phi\right\rangle \] On the positive \(z\) axis, \(\rho=\phi=0\). In the \(xy\)-plane, \(\rho=\phi=\dfrac{\pi}{2}\). On the negative \(z\) axis, \(\rho=\phi=\pi\). There is no restriction on \(\theta\). So the parameter ranges are: \[ 0 \le \phi \le \pi \qquad \text{and} \qquad 0 \le \theta \le 2\pi \]

Now you try it:

Parametrize the piece of the hyperbolic paraboloid \(z=x^2-y^2\) within the cylinder \(x^2+y^2=25\). Give the range of the parameters. The cylinder is shown in blue and the hyperbolic paraboloid is shown in red.

Start with cylindrical coordinates and express \(z\) in turns of \(r\).

The parameterization is: \(\vec R(r,\theta) =\left\langle r\cos\theta,r\sin\theta,r^2(\cos^2\theta-\sin^2\theta\right\rangle\)
and the parameter ranges are: \(0 \le r \le 5\) and \(0 \le \theta \le 2\pi\)

We start with cylindrical coordinates: \[ \vec R(r,\theta,z)=\left\langle r\cos\theta,r\sin\theta,z\right\rangle \] In cylindrical coordinates, the hyperbolic paraboloid is \[ z=x^2-y^2=r^2(\cos^2\theta-\sin^2\theta) \] So the parametrization of the hyperbolic paraboloid becomes \[ \vec R(r,\theta) =\left\langle r\cos\theta,r\sin\theta,r^2(\cos^2\theta-\sin^2\theta)\right\rangle \] The piece of the surface is within the cylinder \(x^2+y^2=25\) or \(r^2=25\) or \(r=5\). So the parameter ranges are: \[ 0 \le r \le 5 \qquad \text{and} \qquad 0 \le \theta \le 2\pi \]

Parametrize the piece of the cylinder \(x^2+z^2=4\) between \(y=0\) and \(y=6+2x+3z\). Give the range of the parameters. The \(y=0\) plane is shown in yellow and the \(y=6+2x+3z\) plane is shown in green.
PY: less lines on yellow plane.

The parameterization is: \(\vec R(\theta,y)=\left\langle 2\cos\theta,y,2\sin\theta\right\rangle\)
and the parameter ranges are: \(0 \le \theta \le 2\pi\) and \(0 \le y \le 6+4\cos\theta+6\sin\theta\)

Since the cylinder is along the \(y\)-axis, we start with cylindrical coordinates along the \(y\)-axis: \[ \vec R(r,\theta,y)=\left\langle r\cos\theta,y,r\sin\theta\right\rangle \] where \(r=\sqrt{x^2+z^2}\). Since the equation of the cylinder is \(x^2+z^2=4\) or \(r^2=4\), we plug \(r=2\) into parametrization to get: \[ \vec R(\theta,y)=\left\langle 2\cos\theta,y,2\sin\theta\right\rangle \] The angle, \(\theta\), is unrestricted. To get the \(y\)-ranges, we plug \(x=2\cos\theta\) and \(z=2\sin\theta\) into the boundaries \(y=0\) and \(y=6+2x+3z\). So the ranges are: \[ 0 \le \theta \le 2\pi \qquad \text{and} \qquad 0 \le y \le 6+4\cos\theta+6\sin\theta \]