MATH 251 – LECTURE 29
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.5–7
webAssign: 14.5–6, due 4/18 11:55 p.m.
Next week: 14.7
Friday:
webAssign: 14.7, opens 4/18 12 a.m.
Midterm 3
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Parametrized surfaces
A vector function r(u, v) = hx(u, v), y(u, v) z(u, v)i, defined on some domain D in the uv-plane, is also said
to ba a parametrization (or a parametric representation) of its image surface S.
Exercise 1. Find a parametric representation of the part of the elliptic paraboloid
z = 6 − 3x2 − 2z 2
that lies above the xy-plane.
Parametrized surfaces
Exercise 1. Find a parametric representation of the part of the elliptic paraboloid
z = 6 − 3x2 − 2z 2
that lies above the xy-plane.
Parametrized surfaces
Exercise 1. Find a parametrization of the surface obtained by rotating the curve x = y + 3y 2, where 1 ≤ y ≤ 2,
around the y-axis.
Parametrized surfaces
Exercise 1. Compute the normal vector r(u, v) of the surface parametrized by
r(u, v) = hu + v, u2, v 2i.
Surface area
Let S be a parametrized surface given by a vector function r(u, v) for (u, v) in some domain D.
Let Rij have side lengths ∆ui and ∆vj . Then the are of Sij is approximately
|(∆ui r0u) × (∆vj r0v )| = |r0u × r0v | ∆ui ∆vj
Surface area
So we can form a Riemann sum
X
|r0u × r0v | ∆ui ∆vj .
P
Definition 2. Let S be a smooth surface given by the vector function r(u, v), which is covered once as (u, v)
ranges through the parameter domain D. Then, the area of the surface S is
ZZ
X
|r0u × r0v | dA.
lim
|r0u × r0v | ∆ui ∆vj =
|P |→0
P
D
Exercise 3. Let D be a region in the xy-plane, viewed in 3-dimensional space. Compute the area of D using
the above formula.
Surface area
Exercise 4. Find the surface area of the part of the surface z = x + y 2 that lies above the triangle with vertices
(0, 0), (1, 1), and (0, 1).
Surface area
Exercise 5. Find the surface area of the sphere with radius R.
Surface area
Exercise 6. Find the surface area of the part of the plane x+2y +3z = 1 that lies inside the cylinder x2 +y 2 = 2.
Surface integrals
Let S be a parametrized surface. It is natural to write
ZZ
Area(S) =
dS,
S
so that
ZZ
ZZ
|r0u × r0v | dA.
dS =
S
D
Definition 7. The integral of a function f (x, y, z) over the parametrized surface S is
ZZ
ZZ
f (x, y, z)dS =
f (r(u, v)) |r0u × r0v | dA.
S
D
Surface integrals
Exercise 8. Evaluate
RR
S
ydS, where S is the part of the plane 3x + 2y + z = 6 that lies in the positive octant.
Surface integrals
Exercise 9. Evaluate
RR p
S
1 + x2 + y 2 dS, where S is parametrized by the vector function
r(u, v) = hu cos(v), u sin(v), vi
for {0 ≤ u ≤ 1, 0 ≤ v ≤ π}.
Surface integrals
Exercise 10. Evaluate S be the unit sphere with density ρ(x, y, z) = z 2 + z + 1. Compute the total mass of S.