Surface areas of parameterized surfaces in MAPLE

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Problem: Find the surface area of the surface given parametrically by

x=3sin(u)cos(v), y=3sin(u)sin(v), z=3cos(v),

where 0<u<2Pi, 0<v<Pi.

> x:=(u,v)->3*sin(u)*cos(v):
x(u,v);
y:=(u,v)->3*sin(u)*sin(v):
y(u,v);
z:=(u,v)->3*cos(v):
z(u,v);

This surface is a "twisted ribbon":

> plot3d([x(u,v),y(u,v),z(u,v)],u=0..2*Pi,v=0..Pi);

The formula for the surface area is

area = integral over u integral over v ||r_u x r_v|| dv du

So, we need to find r_u, r_v, and then the cross product r_u x r_v:

> r:=(u,v)->[x(u,v),y(u,v),z(u,v)]:
r(u,v);

> r_u:=(u,v)->subs({s=u,t=v},diff(r(s,t),s)):
r_u(u,v);
r_v:=(u,v)->subs({s=u,t=v},diff(r(s,t),t)):
r_v(u,v);

> r_u_cross_r_v:=(u,v)->crossprod(r_u(u,v),r_v(u,v)):
r_u_cross_r_v(u,v);

Next, we need the length ||r_u x r_v|| (this "stretch factor" is the area of the parallelogram spanned by the tangent vectors r_u and r_v on the surface):

> F0:=sqrt(dotprod(r_u_cross_r_v(u,v),r_u_cross_r_v(u,v),`orthogonal`));
F:=simplify(F0);

> Int(Int(F,u=0..2*Pi),v=0..Pi);
int(int(F,u=0..2*Pi),v=0..Pi);

Warning, computation interrupted

Problem: Find the surface area of the surface given parametrically by
x=3sin(u)cos(v), y=3sin(u)sin(v), z=3cos(u),
where 0<u<2Pi, 0<v<Pi.

> x:=(u,v)->3*sin(u)*cos(v):
x(u,v);
y:=(u,v)->3*sin(u)*sin(v):
y(u,v);
z:=(u,v)->3*cos(u):
z(u,v);

This is a sphere:

> plot3d([x(u,v),y(u,v),z(u,v)],u=0..2*Pi,v=0..Pi);

The rest is similar to the previous problem:

> r:=(u,v)->[x(u,v),y(u,v),z(u,v)]:
r(u,v);
r_u:=(u,v)->subs({s=u,t=v},diff(r(s,t),s)):
r_u(u,v);
r_v:=(u,v)->subs({s=u,t=v},diff(r(s,t),t)):
r_v(u,v);
r_u_cross_r_v:=(u,v)->crossprod(r_u(u,v),r_v(u,v)):
r_u_cross_r_v(u,v);
F0:=sqrt(dotprod(r_u_cross_r_v(u,v),r_u_cross_r_v(u,v),`orthogonal`));
F:=simplify(F0);

> Int(Int(F,u=0..2*Pi),v=0..Pi);
int(int(F,u=0..2*Pi),v=0..Pi);

>

Problem: Find the surface area of the surface given parametrically by
x=u^2cos(v), y=u^2sin(v), z=u^4,
where 0<u<1, 0<v<Pi

> x:=(u,v)->u^2*cos(v):
x(u,v);
y:=(u,v)->u^2*sin(v):
y(u,v);
z:=(u,v)->u^4:
z(u,v);

This is shaped like a half-bowl:

> plot3d([x(u,v),y(u,v),z(u,v)],u=0..1,v=0..Pi,scaling=constrained);

The rest is similar to the previous problem:

> r:=(u,v)->[x(u,v),y(u,v),z(u,v)]:
r(u,v);
r_u:=(u,v)->subs({s=u,t=v},diff(r(s,t),s)):
r_u(u,v);
r_v:=(u,v)->subs({s=u,t=v},diff(r(s,t),t)):
r_v(u,v);
r_u_cross_r_v:=(u,v)->crossprod(r_u(u,v),r_v(u,v)):
r_u_cross_r_v(u,v);
F0:=sqrt(dotprod(r_u_cross_r_v(u,v),r_u_cross_r_v(u,v),`orthogonal`));
F:=simplify(F0);

> Int(Int(F,u=0..1),v=0..Pi);
int(int(F,u=0..1),v=0..Pi);

>