in (12) and (13), [partial derivative]x([t.sub.i], [theta], [tau], a)/[partial derivative]a and [[partial derivative].sup.2]x([t.sub.i], [theta], [tau], a)/[partial derivative][a.sup.2] are, respectively, the first and second
partial derivative of x(t, [theta], [tau], a) with respect to a and then evaluated at t = [t.sub.i] for i = 0,1,2, ..., n.
A
partial derivative of the component [F.sub.X] (X, Y, Z) over variable X, for example, should be obtained through differentiation of the composite function:
Such a derivative is called a first
partial derivative of u with respect to the single variable.
In Equation A12 the condition is weaker, since a sufficient condition for [partial derivative][x.sup.*.sub.2]/[partial derivative][A.sub.1] < 0 is that the second-order cross
partial derivative of the success function is nonnegative.
[Laplacian operator] J = (1/r.sup.2) times (
partial derivative)/(
partial derivative of r) times (r.sup.2 times (
partial derivative of J/
partial derivative of r)) + (1/r.sup.2 times sin theta) times (partial derivative/partial derivative of theta) times (sin of theta times (partial derive of J /
partial derivative of theta)) + (1/ r.sup.2 times sin.sup.2 of theta) times (
partial derivative squared of J /
partial derivative of phi.sup.2).
First case [[rho].sub.3] [??] [[rho].sub.2]: In the case of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the sign of the
partial derivative [partial derivative]([r.sup.*.sub.L] - [r.sup.*.sub.D])/[partial derivative][alpha] is presented in Table 3.
Taking the
partial derivative of this composition with respect to s at s = 0, we get, using the chain rule,
where the dot denotes the
partial derivative with respect to time.
For a function u (x): [R.sup.n][right arrow] R, we denote the first order
partial derivative by [[partial derivative]u(x)/[partial derivative][x.sub.i]] (i = 1, ..., n) and [[integral].sub.D] u (x) dx the n--fold integral [[integral].sub.a1.sup.b1] ...
Taking the
partial derivative with respect to time we obtain