Functions in differential geometry Maple package atlas

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Features List & Examples

atlasWizard - Maplet™

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 declares functions. In the atlas package a function is non-constant 0-form which depends on other 0-forms.
There are two different syntax of function declaration.
Use first one f=f(x,y,z) to declare function f which depends on x, y, and z;
Use second one F=F(x[i]) i.e. to declare function F which depends on if the working dimension is numeric or if the dimension is symbolic (see atlas[dim] ).
The function identifier can be either symbol or indexed value (see examples below).

Examples:
restart:
with(atlas):

Declare as a constant:
Constants(alpha[k]);

${Catalan, I, _Z, Pi, -I, alpha[k]}$

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]));

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

Verify that h is a function:
'd(h)'=d(h);

Another example:
'd(z)'=d(z);

Some more examples
'd(F)'=d(F);

d(F) = Sum(Diff(F,x[l[1]])*d(x[l[1]]),l[1] = 1 .. dim)

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

d(f*F) = (Diff(f,x)*d(x)+Diff(f,y)*d(y))*F+f*Sum(Diff(F,x[l[2]])*d(x[l[2]]),l[2] = 1 .. dim)

As then:
'd(H[j])'=d(H[j]);

d(H[j]) = Diff(H[j],x)*d(x)+Diff(H[j],y)*d(y)+Sum(Diff(H[j],x[l[1]])*d(x[l[1]]),l[1] = 1 .. dim)+Sum(Diff(H[j],y[l[1]])*d(y[l[1]]),l[1] = 1 .. dim)

But and are just 0-forms:
'd(H[i,j])'=d(H[i,j]);
'd(H)'=d(H);

As then:
'd(alpha[n,m])'=d(alpha[n,m]);

d(alpha[n,m]) = Sum(Diff(alpha[n,m],x[l[1]])*d(x[l[1]]),l[1] = 1 .. dim)

As is a constant then:
'd(alpha[i])'=d(alpha[i]);

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

And more - using exterior product operator (see atlas[`&^`] ):
'd(f)&^d(phi)'=d(f)&^d(phi);

Some trivial examples:
'd(exp(-x^2)+cos(y))'=d(exp(-x^2)+cos(y));

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

And not so trivial ones ( n - was not declared as a constant! ):
'd(f^n)'=d(f^n);

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

${Catalan, I, _Z, Pi, lambda, -I, alpha[k]}$

Thus:
'd(f^lambda)'=d(f^lambda);

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();

$PIECEWISE([{}, Domains],[{}, Mappings],[{}, Tensors],[{}, Forms],[{Catalan, I, _Z, Pi, lambda, -I, alpha[k]}, Constants],[{z, h, f, F, H[i], alpha[i,j]}, Functions])$