With the help of Chain rule and these Jacobian determinants, the relationships between the

partial derivatives of x-y and [xi]-[eta] can be built as follows:

Now, it is straightforward to calculate the

partial derivative of the demand ratio

2]: In the case of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the sign of the

partial derivative [

partial derivative]([r.

A similar procedure is valid for differentiation over any space variable and, hence, either

partial derivative of the field in the laboratory frame coincides with the corresponding

partial derivative in the moving frame: such a derivative similarity takes place.

Taking the

partial derivative of this composition with respect to s at s = 0, we get, using the chain rule,

The following section describes the algorithms for computation of

partial derivatives from the data.

The first-order conditions (FOC) of maximum existence for the function L require that all first-order

partial derivatives of the function with respect to all variables l, c and [epsilon] and to the multiplier [lambda] equal zero.

Note that the average of the two

partial derivatives is (85.

1] < 0 is that the second-order cross

partial derivative of the success function is nonnegative.

With these

partial derivatives, the Jacobian determinant

i] > 0--which includes the case of a monetary effort--because the sign of the

partial derivatives of [e.

In Section 2, we present some results on such subspaces in the particular cases of two points and some or all

partial derivatives of the first order.