atlas[Functions] - declaration of functions 

Calling Sequence: 

    Functions(F1=F1(x1, x2, ...,xn), F2=F2(y1, y2, ..., ym), ..., Fk=Fk(z1, z2, ..., zj)) 

Parameters: 

    Fk=Fk(z1, z2, ..., zj) - equations where Fk-function identifier and zj - variables. 

Description: 

• In the atlas package any identifier is treated as 0-form i.e. as non-constant scalar (if it not declared as constant, p-form, tensor etc. (see atlas[types])).

 

• The Functions procedure allows one to declare functions. In the atlas package a function is non-constant 0-form which depends on other 0-forms.

 

• There are two different syntaxes for function declaration.

 

• Use the first form f=f(x,y,z) to declare a function f depending on x, y, and z;

 

• Use the second form F=F(x[i]) i.e. to declare a function F depending on if the working dimension is numeric or if the dimension is symbolic (see atlas[dim]).

 

• The function identifiers can be either symbolic or indexed values (see examples below).

 

Examples: 

>	restart: with(atlas):

 

Declare as a constant: 

>	Constants(alpha[k]);

 

(2.1)

 

Declare as functions:  

>	Functions(f=f(x,y),h=h(phi,psi,chi),z=z(f,h),F=F(x[i]), H[i]=H[i](x,y,x[k],y[j]),alpha[i,j]=alpha[i,j](x[k]));

 

(2.2)

 

Varify that f is a function using exterior derivative operator (see atlas[d]):  

>	'd(f)'=d(f);

 

(2.3)

 

Verify that h is a function: 

>	'd(h)'=d(h);

 

(2.4)

 

Another example:  

>	'd(z)'=d(z);

 

(2.5)

 

Some more examples
'd(F)'=d(F); 

(2.6)

 

>	'd(f*F)'=d(f*F);

 

(2.7)

 

As then: 

>	'd(H[j])'=d(H[j]);

 

(2.8)

 

But and are just 0-forms: 

>	'd(H[i,j])'=d(H[i,j]); 'd(H)'=d(H);

 

 

	(2.9)

 

As  then: 

>	'd(alpha[n,m])'=d(alpha[n,m]);

 

(2.10)

 

As is a constant then: 

>	'd(alpha[i])'=d(alpha[i]);

 

(2.11)

 

But: 

>	'd(alpha[i,j,k])'=d(alpha[i,j,k]);

 

(2.12)

 

And more - using exterior product operator (see atlas[`&^`]): 

>	'd(f)&^d(phi)'=d(f)&^d(phi);

 

(2.13)

 

Some trivial examples: 

>	'd(exp(-x^2)+cos(y))'=d(exp(-x^2)+cos(y));

 

(2.14)

 

>	'd(f*x^3)'=d(f*x^3);

 

(2.15)

 

And not so trivial ones (n - was not declared as a constant!): 

>	'd(f^n)'=d(f^n);

 

(2.16)

 

Declare lambda as a constant (see atlas[Constants]):
Constants(lambda); 

(2.17)

 

Thus:
'd(f^lambda)'=d(f^lambda); 

(2.18)

 

Let's see "who is who" 

>	Who([f,h,x,y,z,lambda]);

 

f: function h: function x: 0 - form y: 0 - form z: function lambda: constant

 

>

Who();

 

(2.19)

 

>

 

See Also:  

atlas, atlas[Constants], atlas[Forms], atlas[Vectors], atlas[Tensors], atlas[d], atlas[`&^`], atlas[Who].