1.8: Surfaces

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Oct 27, 2024
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- 1.7: Tangent Planes and Normal Lines
- 1.9: Partial Derivatives

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Cylinders

Let C be a curve, then we define a cylinder to be the set of all lines through C and perpendicular to the plane that C lies in.

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We can tell that an equation is a cylinder is it is missing one of the variables.

Quadric Surfaces

Recall that the quadrics or conics are lines , hyperbolas, parabolas, circles, and ellipses. In three dimensions, we can combine any two of these and make a quadric surface. For example

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is a paraboloid since for constant z we get a circle and for constant x or y we get a parabola. We use the suffix -oid to mean ellipse or circle. We have:

$\begin{align*} \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2 } + \dfrac{z^2}{c^2} &=1 \;\;\; \text{is an ellipsoid,} \\ -\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2 } + \dfrac{z^2}{c^2} &=1 \;\;\; \text{is a hyperboloid of 2 sheets, while} \\ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2 } - \dfrac{z^2}{c^2} &=1 \;\;\; \text{is a hyperboloid of 1 sheet.} \end{align*} \nonumber$

Surface of Revolution

Let $y = f(x)$ be a curve, then the equation of the surface of revolution abut the x-axis is

$y^2 + z^2 = f^2(x). \nonumber$

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Example $\PageIndex{1}$

Find the equation of the surface that is formed when the curve

$y = \sin x\nonumber$

with

$0 \le x < \dfrac{\pi}{2} \nonumber$

is revolved around the y-axis.

Solution

This uses a different formula since this time the curve is revolved around the y-axis.

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The circular cross section has radius $\sin^{-1}y$ and the circle is perpendicular to the y-axis. Hence the equation is

$x^2 + z^2 = (\sin^{-1} y)^2. \nonumber$

Cylindrical Coordinates

We can extend polar coordinates to three dimensions by

$x = r\, \cos\, q, \nonumber$

$y = r\, \sin\, q,\nonumber$

$z = z.\nonumber$

Example $\PageIndex{2}$

We can write $(1,1,3)$ in cylindrical coordinates. We find

$r = \sqrt{1^2 + 1^2} = sqrt{2} \nonumber$

and

$\theta = \tan^{-1} \left(\dfrac{1}{1} \right) = \dfrac{\pi}{4} \nonumber$

so that the cylindrical coordinates are

$( \sqrt{2}, \frac{\pi}{4},3).\nonumber$

Spherical Coordinates

An alternate coordinate system works on a distance and two angle method called spherical coordinates. We let $r$ denote the distance from the point to the origin, $\theta$ represent the same $\theta$ as in cylindrical coordinates, and $\phi$ denote the angle from the positive z-axis to the point.

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The picture tells us that

$r = r \sin \phi\nonumber$

and that

$z = r \cos \phi .\nonumber$

From this we can find

$x = r\cos \theta = r \sin \phi \cos \theta$ ,
$y = r \sin \theta = r \sin \phi \sin \theta$ ,
$z = r \cos \phi$ .

Immediately we see that

$x^2 + y^2 + z^2 = r^2. \nonumber$

We use spherical coordinates whenever the problem involves a distance from a source.

Example $\PageIndex{3}$

Convert the surface

$z = x^2 + y^2 \nonumber$

to an equation in spherical coordinates.

Solution

We add $z^2$ to both sides

$z + z^2 = x^2 + y^2 + z^2. \nonumber$

Now it is easier to convert

$r\, \cos \phi + r^2 \, \cos^2 \phi = r^2. \nonumber$

Divide by $r$ to get

$\cos \phi + r\, \cos^2 \phi = r .\nonumber$

Now solve for $r$ .

$r = \dfrac{\cos \phi }{1 - \cos^2 \phi } = \dfrac{\cos \phi }{\sin^2 \phi } = \csc \phi \; \cot \phi .\nonumber$

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Example $\PageIndex{1}$

Solution

Example $\PageIndex{2}$

Example $\PageIndex{3}$

Solution

Cylinders

Quadric Surfaces

Surface of Revolution

Example 1.8.1\PageIndex{1}

Solution

Cylindrical Coordinates

Example 1.8.2\PageIndex{2}

Spherical Coordinates

Example 1.8.3\PageIndex{3}

Solution

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Example $\PageIndex{1}$

Example $\PageIndex{2}$

Example $\PageIndex{3}$