Surfaces and Transformations
Summary and Review
In this chapter, we learned to use derivatives and vectors together in
order to study the geometry of surfaces, coordinate systems, and
coordinate transformations. We began by showing that the gradient
of a level surface is normal to the surface. Often,
however, surfaces are studied parametrically, where a parameterization
of a surface is a function of the form
r( u,v) =
á x( u,v) ,y(u,v) ,z( u,v)
ñ 

The principal vectors r_{u} =
áx_{u},y_{u},z_{u}
ñ and r_{v} =
áx_{v},y_{v},z_{v}
ñ are tangent to the surface, thus allowing
us to parameterize the tangent plane at a given point r(p,q) on the surface by
L( du,dv) = r( p,q) +r_{u}( p,q) du+r_{v}( p,q) dv 

Also, the cross product r_{u}×r_{v} is normal to
the surface, thus allowing us to define the unit surface normal n to an oriented surface to be the unit vector in the direction of the normal
to the surface at each point.
Parameterizations in which z( u,v) = 0 for all u and v are
called coordinate transformations and are of the form
T( u,v) =
á x( u,v) ,y( u,v)
ñ 

The derivative of a coordinate transformation is called the Jacobian of the
transformation and is defined
As a transformation, the Jacobian J( u,v) maps tangent vectors
to a curve C in the uvplane to tangent vectors to the image of C in
the xyplane. It follows that the determinant of the transformation
closely scales the area of a small region in the uvplane to the area of
the image of the region in the xyplane. We capture this idea with the
area differential, which is given by
dA = 
ê ê


¶( x,y)
¶( u,v)

ê ê

dudv 

Jacobians, normals, parameterizations, and coordinate transformations are
instrumental in the study of coordinate systems other than the usual
Euclidean coordinate systems. In the plane, polar coordinates are often used
to locate points, where polar coordinates are defined by the relationships
which immediately imply that x^{2}+y^{2} = r^{2}. If polar coordinates are
applied to the xycoordinates in 3 dimensional space, then the result is
known as cylindrical coordinates. Also, spherical coordinates form an alternative
coordinate system in 3dimensional space and are defined by 2 sets of
relationships:
These lead to the fundamental idea x^{2}+y^{2}+z^{2} = r^{2}, and once
combined, lead also to the more direct expression of the Spherical
coordinate transformation:
Finally, given a parametric surface r( u,v) , we
define g_{11} = r_{u}·r_{u}, g_{12} = r_{u}·r_{v}, and g_{22} = r_{v}·r_{v}.
It then follows that short distances ds on the surface satisfy
ds^{2} = g_{11}du^{2}+2g_{12}dudv+g_{22}dv^{2} 

which is known as the fundamental form of the surface. If g_{12} = r_{u}·r_{v} = 0, then the parameterization is said to be
orthogonal and the fundamental form resembles the Pythagorean theorem.
Properties of the surface which can be derived from the fundamental form are
said to be intrinsic to the surface. Intrinsic properties include the
length of a curve and geodesics of the surface, which are curves whose
acceleration is normal to the surface at each point on the curve. The
curvature of a geodesic is said to be a normal curvature of a surface, and
the maximum and minimum curvatures at a given point are said to be the
principal curvatures of the surface at that point. The sum of the principal
curvatures is said to be the mean curvature of the surface, and the product
of the curvatures is said to be the Gaussian Curvature of the surface.
Gauss' Theorem Egregium shows that the Gaussian curvature of a surface is
intrinsic to the surface.
There are other ideas and concepts introduced in this chapter, and you should
reread the individual sections in addition to this summary. The
review materials are based on both the ideas above and some of those in
the chapter not mentioned here. Review questions and solutions are
in web page form on the left and in pdf form on the right. For
maximum benefit, you should attempt to answer the questions before
you look at the solutions.
You will need Acrobat reader in order to open and print the pdf files.
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