Partial Differentiation of Real Ternary Functions

Partial Differentiation of Real Ternary Functions

For simplicity, we adopt the following convention: D is a set, x, x0, y, y0, z, z0, r, s, t are real numbers, p, a, u, u0 are elements of R3, f , f1, f2, f3, g are partial functions from R3 to R, R is a rest, and L is a linear function. We now state three propositions: (1) dom proj(1, 3) = R3 and rng proj(1, 3) = R and for all elements x, y, z of R holds (proj(1, 3))(〈x, y, z〉) = x. (2) dom pro...

Second-Order Partial Differentiation of Real Ternary Functions

For simplicity, we adopt the following rules: x, x0, y, y0, z, z0, r denote real numbers, u, u0 denote elements of R3, f , f1, f2 denote partial functions from R3 to R, R denotes a rest, and L denotes a linear function. Let f be a partial function from R3 to R and let u be an element of R3. We say that f is partial differentiable on 1st-1st coordinate in u if and only if the condition (Def. 1) ...

Partial Differentiation of Real Binary Functions

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, Z is a subset of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Next we state two propositions: (1) domproj(1, 2) = R2 and rng proj(1, 2) = R and for all elements x, y of R holds (proj(1, 2))(〈x, y〉) = x. (2) domproj(2, 2) = R2 and rng proj(...

Second-Order Partial Differentiation of Real Binary Functions

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Let us note that every rest is total. Let f be a partial function from R2 to R and let z be an element of R2. The functor pdiff1(f, z) yielding a function from R2 into R is defined as follows: (Def....

A characterization of triple semigroup of ternary functions and Demorgan triple semigroup of ternary functions