Chapter 16.4 Review

Question 1

what is the generic formula for a parametrized surface?

Answer 1

the generic formula for parametrized surface is G(u,v)=(x(u,v), y(u,v), z(u,v)) where parameters u and v vary in a domain D in the uv-plane.

Question 2

What is the formula for Tangent vector?

Answer 2

Tu=dG/du=<dx></dx>

Question 3

What is the formula for normal vector?

Answer 3

Tv=dG/dv=<dx></dx>

Question 4

What does the formula look like for cross product of normal and tangent vecotr?

Answer 4

n=n(u,v)=T_u X T_v.

Question 5

What is the formula for Area(S) of parametrized surface?

Answer 5

Area(S)=∫∫_D||n(u,v)||dudv

Question 6

What is the formula for a function over a parameterized surface ?

Answer 6

∫∫_Sf(x,y,z)dS=∫∫_Df(G(u,v))||n(u,v)||dudv

Question 7

what is the standard parametrization of cylinder of Radius R?

Function parametrization, outward normal, and dS

Answer 7

A cylinder of Radius R, (central axis on z-axis)

G(Θ,z)=(RcosΘ,RsinΘ,z)

Outward Normal: n=T_ΘxT_z=R<cos></cos>

dS=||n||dΘdz=RdΘdz

Question 8

what is the standard parametrization of sphere of Radius R centred at origin ?

Function parametrization, outward normal, and dS

Answer 8

G(Θ, ϕ)=(RcosΘsinϕ,RsinΘsinϕ,Rcosϕ)

Unit Radial Vector: e_r=<cos></cos>

Outward normal: n=T_ϕX T_{<span>Θ=</span>}(R²sinϕ)e_r

dS=||n||dϕdΘ=R²sinϕdϕdΘ

Question 9

what is the parametrization of the graph of z=g(x,y) along with normal, and dS

Answer 9

Graph of z=g(x,y)

G(x,y)=(x,y,g(x,y))

n=T_x X T_y=<-g_x,-g_y,1>

dS=||n||dxdy=√(1+g_x^2+g_y^2)dxdy