Notes on lecture 18
In our earlier lectures, we had considered areas of plane surfaces or sur-
faces of revolution only. In this lecture, we will study “Parametric surfaces”
and their areas.
1 Parametric surfaces and their areas
1.1 Parametric curves
Recall that in the case of parametric curves, we had considered an interval
[a, b], and t was a variable such that a t b. Then we defined some
functions x(t), y(t), z(t) from [a, b] to the real line R. As the variable t runs
through [a, b], for each value of t, we are going to get a point (x(t), y(t), z(t))
in space. If we plot all these points in space (as t varies in the interval
[a, b]), we are going to get a curve. Corresponding to an arbitrary point
P = (x(t), y(t), z(t)) lying on the curve so generated, we can always define a
vector ~r(t) whose components are x(t), y(t), and z(t).
1.2 Parametric surfaces
In case of parametric curves, we had considered an interval [a, b]. Now, let
us consider a domain D (✓ R2 ) in a 2 dimensional plane (which we call the
uv-plane). Define 3 functions x(u, v), y(u, v), and z(u, v) from D to R. Then
corresponding to each point (u, v) 2 D, we are going to obtain a point in the
3 dimensional space. Now if we consider the union of all such points in the
3 dimensional space, then we are going to obtain a surface S. This surface
is going to be called a parametric surface, because it is generated by the
2 parameters u and v. The equations
x = x(u, v), y = y(u, v), z = z(u, v)
1
,are called the parametric equations for the surface S. So basically, cor-
responding to each point (u, v) 2 D, we are going to get a point P =
(x(u, v), y(u, v), z(u, v)) on the surface S. Now if we joing the origin with
the point P , we are going to get a vector, which we denote by ~r(u, v). Here,
~r(u, v) = x(u, v)î + y(u, v)ĵ + z(u, v)k̂.
Figure 1.2.1 explains this phenomenon.
Figure 1.2.1: A parametric surface being generated
Remark 1.2.1. Earlier in the case of parametric curves, the vector ~r(t) was
defined in terms of a single parameter t, but in this case (of parametric surface)
the vector ~r(u, v) is defined in terms of 2 variables u and v.
Example 1.2.2. Let ~r(u, v) = uî + u cos v ĵ + u sin v k̂. Find the surface given
by the above parametric representation.
Clearly,
~r(u, v) = xî + y ĵ + z k̂,
where x = u, y = u cos v, and z = u sin v.
Observe that y 2 + z 2 = u2 = x2 . So the equation of the surface is given by
x2 = y 2 + z 2 . ⇤
Example 1.2.3. Find a parametric representation for the cylinder
x2 + y 2 = 4, 0 z 1.
2
, A parametric representation is given by
~r(✓, z) = xî + y ĵ + z k̂,
where
x = 2 cos ✓, y = 2 sin ✓, z = z, 0 ✓ 2⇡, 0 z 1.
⇤
Example 1.2.4. Find a vector function that represents the elliptic paraboloid
z = x2 + 2y 2 .
The required vector function is given by
~r(x, y) = xî + y ĵ + (x2 + 2y 2 )k̂.
⇤
Remark 1.2.5. Whenever we have a surface which is the graph of a function of
2 variables, say given by z = f (x, y), then the job of finding out a vector function
that represents the surface is easy (as can be seen in the above example). It is
just given by
~r(x, y) = xî + y ĵ + f (x, y)k̂.
1.3 The grid curves
Now we are going to see how constant u, v lines (that is, lines of the form
u = u0 and v = v0 ) in the region D of the uv-plane will correspond to some
curves over the surface S called “grid curves”.
Consider a domain D in the uv-plane. Let (u0 , v0 ) 2 D. Consider the lines
u = u0 and v = v0 in D. Let S denote the parametric surface generated by
all the points in D via
~r(u, v) = x(u, v)î + y(u, v)ĵ + z(u, v)k̂.
Let us now see how the images of the lines u = u0 and v = v0 in D will look
like in the surface S. Observe that
~r(u0 , v) = x(u0 , v)î + y(u0 , v)ĵ + z(u0 , v)k̂,
which is essentially a function of the single parameter v. A single parameter
function will generate a curve in the 3 dimensional space. So, the line u = u0
3
In our earlier lectures, we had considered areas of plane surfaces or sur-
faces of revolution only. In this lecture, we will study “Parametric surfaces”
and their areas.
1 Parametric surfaces and their areas
1.1 Parametric curves
Recall that in the case of parametric curves, we had considered an interval
[a, b], and t was a variable such that a t b. Then we defined some
functions x(t), y(t), z(t) from [a, b] to the real line R. As the variable t runs
through [a, b], for each value of t, we are going to get a point (x(t), y(t), z(t))
in space. If we plot all these points in space (as t varies in the interval
[a, b]), we are going to get a curve. Corresponding to an arbitrary point
P = (x(t), y(t), z(t)) lying on the curve so generated, we can always define a
vector ~r(t) whose components are x(t), y(t), and z(t).
1.2 Parametric surfaces
In case of parametric curves, we had considered an interval [a, b]. Now, let
us consider a domain D (✓ R2 ) in a 2 dimensional plane (which we call the
uv-plane). Define 3 functions x(u, v), y(u, v), and z(u, v) from D to R. Then
corresponding to each point (u, v) 2 D, we are going to obtain a point in the
3 dimensional space. Now if we consider the union of all such points in the
3 dimensional space, then we are going to obtain a surface S. This surface
is going to be called a parametric surface, because it is generated by the
2 parameters u and v. The equations
x = x(u, v), y = y(u, v), z = z(u, v)
1
,are called the parametric equations for the surface S. So basically, cor-
responding to each point (u, v) 2 D, we are going to get a point P =
(x(u, v), y(u, v), z(u, v)) on the surface S. Now if we joing the origin with
the point P , we are going to get a vector, which we denote by ~r(u, v). Here,
~r(u, v) = x(u, v)î + y(u, v)ĵ + z(u, v)k̂.
Figure 1.2.1 explains this phenomenon.
Figure 1.2.1: A parametric surface being generated
Remark 1.2.1. Earlier in the case of parametric curves, the vector ~r(t) was
defined in terms of a single parameter t, but in this case (of parametric surface)
the vector ~r(u, v) is defined in terms of 2 variables u and v.
Example 1.2.2. Let ~r(u, v) = uî + u cos v ĵ + u sin v k̂. Find the surface given
by the above parametric representation.
Clearly,
~r(u, v) = xî + y ĵ + z k̂,
where x = u, y = u cos v, and z = u sin v.
Observe that y 2 + z 2 = u2 = x2 . So the equation of the surface is given by
x2 = y 2 + z 2 . ⇤
Example 1.2.3. Find a parametric representation for the cylinder
x2 + y 2 = 4, 0 z 1.
2
, A parametric representation is given by
~r(✓, z) = xî + y ĵ + z k̂,
where
x = 2 cos ✓, y = 2 sin ✓, z = z, 0 ✓ 2⇡, 0 z 1.
⇤
Example 1.2.4. Find a vector function that represents the elliptic paraboloid
z = x2 + 2y 2 .
The required vector function is given by
~r(x, y) = xî + y ĵ + (x2 + 2y 2 )k̂.
⇤
Remark 1.2.5. Whenever we have a surface which is the graph of a function of
2 variables, say given by z = f (x, y), then the job of finding out a vector function
that represents the surface is easy (as can be seen in the above example). It is
just given by
~r(x, y) = xî + y ĵ + f (x, y)k̂.
1.3 The grid curves
Now we are going to see how constant u, v lines (that is, lines of the form
u = u0 and v = v0 ) in the region D of the uv-plane will correspond to some
curves over the surface S called “grid curves”.
Consider a domain D in the uv-plane. Let (u0 , v0 ) 2 D. Consider the lines
u = u0 and v = v0 in D. Let S denote the parametric surface generated by
all the points in D via
~r(u, v) = x(u, v)î + y(u, v)ĵ + z(u, v)k̂.
Let us now see how the images of the lines u = u0 and v = v0 in D will look
like in the surface S. Observe that
~r(u0 , v) = x(u0 , v)î + y(u0 , v)ĵ + z(u0 , v)k̂,
which is essentially a function of the single parameter v. A single parameter
function will generate a curve in the 3 dimensional space. So, the line u = u0
3
