VECTOR DIFFERENTIATION
37
are the partial derivatives of A with respect to y and z respectively if these limits exist. The remarks on continuity and differentiability for functions of one variable can be extended to For example, c (x, y) is called continuous at (x, y) if Jim 0 (x +Ax, y +Ay) = q5 (x,y), or if for each positive number e we can find some positive number
functions of two or more variables. AY-0
8 such that 0 (x +Ax, y +Ay) - gb (x,y)1 < E whenever nitions hold for vector functions.
j Ax j
< 8 and
I Ay I
< 8. Similar defi-
For functions of two or more variables we use the term differentiable to mean that the function has continuous first partial derivatives. (The term is used by others in a slightly weaker sense.) Higher derivatives can be defined as in the calculus. Thus, for example, a2A
a A ax (ax) ,
_
axe
a aA = ax(ay
a2A
ax ay
aA ay -6 y)
2A
aye a2A
ay ax
=
a aA
a2A
az(az )
az2
a (aA ay ax
a
a3A
ax az2
If A has continuous partial derivatives of the second order at least, then order of differentiation does not matter.
a2A
- ax az2 a 2 A'
ax ay
- aya 2ax`
,
i.e. the
Rules for partial differentiation of vectors are similar to those used in elementary calculus for scalar functions. Thus if A and B are functions of x,y,z then, for example, 1.
ax (A B) = A - a$ + 2A . B
ax(Ax B) = Ax aB
+
aAx B
{ax(A.B)}
=
=
ay
y
A
a2B
+
ay ax
aA ay
3B +
ax
{A.aB
+
aA aB ax
ay
aA.B} +
a2A .B ay ax '
etc.
DIFFERENTIALS OF VECTORS follow rules similar to those of elementary calculus. For example, 1.
If A = Al' + A2j + A3k , then dA = dA1i + dA2j + dA3k
2. d(A B) = A dB + dA B
3. d(AxB) = AxdB + dAxB 4.
If A = A(x,y,z), then dA = a A dx + aA dy + a A dz ,
etc.
DIFFERENTIAL GEOMETRY involves a study of space curves and surfaces. If C is a space curve defined by the function r(u), then we have seen that du is a vector in the direction of the tangent to C. If the scalar u is taken as the arc length s measured from some fixed point on C, then -d-r- is a unit tangent vector to C and is denoted by T (see diagram below). The