Defining families of function from $\mathbb {R}$^p-1 to $\mathbb {R}$^q using a function from $\mathbb {R}$^p to $\mathbb {R}$^q

Suppose that the function f : (x, y) $\rightarrow$ f (x, y) is defined, and we want to define a family of functions g(t) such that g(t)(y) : = f (t, y) (i.e. t is viewed as a parameter). Since the expression after -> (or :=) is not evaluated, we should not define g(t) by g(t):=y->f(t,y), we have to use the unapply command.

For example, assuming that f : (x, y) $\rightarrow$ x sin(y) and g(t) : y $\rightarrow$ f (t, y), input :

f(x,y):=x*sin(y);g(t):=unapply(f(t,y),y)

Output :

((x,y)->x*sin(y), (t)->unapply(f(t,y),y))

Input

g(2)

Output :

y->2 ^. sin(y)

Input

g(2)(1)

Output :

2 ^. sin(1)

Next example, suppose that the function h : (x, y) $\rightarrow$ [x*cos(y), x*sin(y)] is defined, and we want to define the family of functions k(t) having t as parameter such that k(t)(y) : = h(t, y). To define the function h(x, y), input :

h(x,y):=(x*cos(y),x*sin(y))

To define properly the function k(t), input :

k(t):=unapply(h(x,t),x)

Output :

(t)->unapply(h(x,t),x)

Input

k(2)

Output :

(x)->(x*cos(2),x*sin(2))

Input

k(2)(1)

Output :

(2*cos(1),2*sin(1))