Guide to LaTeX (4th Edition) (Tools and Techniques

150 Chapter 5. Mathematical Formulas
Hint: recall that @{...} expressions in the formatting field of the array environment
can be used to insert horizontal spacing and/or mathematical text between
the columns (see the first example in Section 5.4.3).
Exercise 5.21: Create the following output with the array environment.
Equations for the tangential plane and surface normal
Equation
for the Tangential plane Surface normal
surface
F(x, y, z) = 0
∂F
∂F
(X − x) + (Y − y)
∂x ∂y
+ ∂F
(Z − z) = 0
∂z
z = f (x, y) Z − z = p(X − x) + q(Y − y)
x = x(u, v)
y = y(u, v)
z = z(u, v)
r = r(u, v)
X − x Y − y Z − z
∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
∂z
∂u
∂z
∂v
(R − r)(r1 × r2) = 0
or (R − r)N = 0
= 0
X − x
∂F
∂x
= Y − y
∂F
∂y
= Z − z
∂F
∂z
X − x Y − y
=
p q
Z − z
=
−1
X − x
∂y ∂z
∂u ∂u
=
Y − y
∂z ∂x
∂u ∂u
∂y
∂v
∂z
∂v
∂z
∂v
∂x
∂v
= Z − z
∂x ∂y
∂u ∂u
∂x
∂v
∂y
∂v
R = r + λ(r1 × r2)
or R = r + λN
In this Table x, y, z and r are the coordinates and the radius vector of a fixed
point M on the curve; X, Y , Z and R are the coordinates and radius vector
of a point on the tangential plane or surface normal with reference to M;
furthermore p = ∂z
∂x
, q = ∂z
∂y and r1 = ∂r/∂u, r2 = ∂r/∂v.
Note: If you succeed in reproducing this mathematical table you should have no
more problems with positioning formulas and their parts!
Hints for the solution:
1. Define abbreviations such as \D for \displaystyle and \bm for \boldmath
and possibly even \ba and \ea for \begin{array} and \end{array}.
2. Build the table in stages. Start with the head and only proceed further
when it looks reasonable. Watch out that normal text within the array
environment must be included inside an \mbox.
3. Continue with the first mathematical row. Here the first entry in the second
column is also an array environment that is aligned with the rest of the
row by means of the [t] positioning argument. Do not forget to activate
the size \D at the necessary places in the inner structures. Insert a possible
strut (Section 4.7.6) to generate the right distance from the head. The
distance to the next row can be adjusted with a length specification added
to the row terminating command \\[..].