Functions, Sets, Gradient 127/ are surfaces. This observation leads to several possible ways to define
curves and surfaces.
Consider the space R^2 with cartesian coordinates (x,y). Then each of
the following defines a curve in R^2 :
- A graph y — g(x) of a function of one variable.
- A level set f(x, y) = c of a function of two variables.
- A vector-valued function r(t) = x(t) i+y(t)j, to <t < t\, of one variable.
EXERCISE 3.1.7. (a)B Give an example of a curve that cannot be repre-
sented as a graph y = g(x). (b)A+ Is there a continuous curve that cannot
be represented as a level set of a function of two variables?
Consider the space R^3 with cartesian coordinates (x,y,z). Then each
of the following defines a surface in R^3 :
- A graph z = f{x,y) of a function of two variables;
- A level set F(x,y,z) = c of a function of three variables;
- A a vector-valued function r(u, v) = x(u, v)i + y(u, v) j+ z(u, v) k. of two
variables; this is known as a parametric representation of the surface.
EXERCISE 3.1.8. (a)B Give an example of a surface that cannot be rep-
resented as a graph z = f(x,y). (b)B Think of at least two different ways
to represent a curve in R^3. (c)A+ Is there a surface that can be repre-
sented parametrically using a continuous function r = r(u,v) but cannot
be represented as a level set of a function of three variables?
EXERCISE 3.1.9. Below, we summarize some geometric properties of the
gradient that are usually studied in a vector calculus class.
(a) Let f(x,y) = c be a curve. Let {xo,yo) be a point on the curve and
assume that the function f is differentiable at (xo,yo)- (i) Show that the
curve is smooth at the point (xo,yo) if and only if V/(xo, j/o) ¥" 0. (ii)
Use (3.1.9) to show that V/(:ro,2/o) is perpendicular to the unit tangent
vector of the curve at the point (xo,yo)- Hint: in this case, g(t) = c and so
g'(to) = 0. (Hi) Write the equation of the normal line to the curve at the
point (xo,yo). Hint: the line has V/(xo,j/o) as the direction vector.
(b) Let F(x, y,z) = c be a surface. Let Po = (xo, yo, ZQ) be a point on the
surface, and C, a smooth curve on the surface passing through the point Po.
Assume that the function F is differentiable at PQ. (i) Show that 'VF(PQ)
is perpendicular to the unit tangent vector to C at PQ. Hint: same arguments
as in part (a), (ii) Write the equation of the tangent plane to the surface
at the point PQ. Hint: the plane has Vir(Fo) as the normal vector.