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T ∂u
u = c1
v = c2
r(u, v)
Figure 10.4 The tangent plane T to a surface S at a particular point P ;
u = c1 and v = c2 are the coordinate curves, shown by dotted lines, that pass
through P . The broken line shows some particular parametric curve r = r(λ)
lying in the surface.
10.5 Surfaces
A surface S in space can be described by the vector r(u, v) joining the origin O of
a coordinate system to a point on the surface (see ﬁgure 10.4). As the parameters
u and v vary, the end-point of the vector moves over the surface. This is very
similar to the parametric representation r(u) of a curve, discussed in section 10.3,
but with the important diﬀerence that we require two parameters to describe a
surface, whereas we need only one to describe a curve.
In Cartesian coordinates the surface is given by
r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k,
where x = x(u, v), y = y(u, v) and z = z(u, v) are the parametric equations of the
surface. We can also represent a surface by z = f(x, y) or g(x, y, z) = 0. Either
of these representations can be converted into the parametric form in a similar
manner to that used for equations of curves. For example, if z = f(x, y) then by
setting u = x and v = y the surface can be represented in parametric form by
r(u, v) = ui + vj + f(u, v)k.
Any curve r(λ), where λ is a parameter, on the surface S can be represented
by a pair of equations relating the parameters u and v, for example u = f(λ)
and v = g(λ). A parametric representation of the curve can easily be found by
straightforward substitution, i.e. r(λ) = r(u(λ), v(λ)). Using (10.17) for the case
where the vector is a function of a single variable λ so that the LHS becomes a
total derivative, the tangent to the curve r(λ) at any point is given by
∂r du ∂r dv
∂u dλ ∂v dλ
The two curves u = constant and v = constant passing through any point P
on S are called coordinate curves. For the curve u = constant, for example, we
have du/dλ = 0, and so from (10.21) its tangent vector is in the direction ∂r/∂v.
Similarly, the tangent vector to the curve v = constant is in the direction ∂r/∂u.
If the surface is smooth then at any point P on S the vectors ∂r/∂u and
∂r/∂v are linearly independent and deﬁne the tangent plane T at the point P (see
ﬁgure 10.4). A vector normal to the surface at P is given by
× .
∂u ∂v
In the neighbourhood of P , an inﬁnitesimal vector displacement dr is written
dr =
du +
The element of area at P , an inﬁnitesimal parallelogram whose sides are the
coordinate curves, has magnitude
∂r ∂r
∂r dS = du ×
dv = × du dv = |n| du dv.
∂u ∂v
Thus the total area of the surface is
× ∂r du dv =
|n| du dv,
∂v R ∂u
where R is the region in the uv-plane corresponding to the range of parameter
values that deﬁne the surface.
Find the element of area on the surface of a sphere of radius a, and hence calculate the
total surface area of the sphere.
We can represent a point r on the surface of the sphere in terms of the two parameters θ
and φ:
r(θ, φ) = a sin θ cos φ i + a sin θ sin φ j + a cos θ k,
where θ and φ are the polar and azimuthal angles respectively. At any point P , vectors
tangent to the coordinate curves θ = constant and φ = constant are
= a cos θ cos φ i + a cos θ sin φ j − a sin θ k,
= −a sin θ sin φ i + a sin θ cos φ j.
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