/*MATH214.mac 
Contents for use with help() */
disp("MATH214.mac:  help(MATH214) for contents");

MATH214_help():=block(printf(true,
"MATH214.mac contains:
    cross(u,v)
    dot(u,v)
    len(u)
    curvature(r,t)
    unitT(r,t)
    unitN(r,t)
    unitB(r,t)
    integratePaths(f,r,t,a,b)
    integratePathv2(F,r,t,a,b)
    integratePathv3(F,r,t,a,b)
    curl(F,x,y,z)
    div(F,x,y,z)
    grad(F,x,y,z)
	-
	-
    for any of the above functions,
    help(function_name) returns help lines for function_name"
)
);
/*
The help utility
*/
help(_input):=block(
eval_string(sconcat(string(_input),"_help()")),
return(done)
);



/* cross computes the cross produce of two vectors of dimension 3 */
cross(_u,_v):=[_u[2]*_v[3]-_u[3]*_v[2],_u[3]*_v[1]-_u[1]*_v[3],_u[1]*_v[2]-_u[2]*_v[1]]$
/*the help file for use with help()*/
cross_help():=block(printf(true,
"cross(u,v) computes the cross produce of two vectors of dimension 3
     Example
        cross([1,0,0],[0,1,0]); returns
           [0,0,1]"
)
);

/* dot duplicates the . operator for parallelism with cross */
dot(_u,_v) :=_u._v$
/*the help file for use with help()*/
dot_help():=block(printf(true,
"dot(u,v) duplicates the Maxima . operator for function-name 
  parallelism with cross(u,v)
     Example
        dot([1,1,0],[0,1,1]); returns
           1 "
)
);

/* len computes the vector length */
len(_u):= sqrt(_u._u)$
/*the help file for use with help()*/
len_help():=block(printf(true,
"len(u,v) len computes the vector length
     Example
        len([0,1,1]); returns
           sqrt(2) "
)
);



/* curvature computes the curvature  of a vector values function 
    r: R --> R^2 or r: R --> R^3
   curvature requires unitT  */
curvature(r,t):=
block([T,Tdot,rdot,K],
    define(T(t),unitT(r,t)),
	define(Tdot(t),diff(T(t),t)),
	define(rdot(t),diff(r,t)),
	define(K(t),trigsimp(sqrt(Tdot(t).Tdot(t)/rdot(t).rdot(t)))),
	K(t)
)$
/*the help file for use with help()*/
curvature_help():=block(printf(true,
"curvature computes the curvature  of a vector values function 
 r: R --> R^2 or r: R --> R^3
      Example
          curvature(2*[cos(t),sin(t)],t); returns
             1/2 "
)
);

/* unitT computes the unit tangent vector for a vector valued function
 r: R --> R^2 or   r: R --> R^3
of a scalar variable r(t)=[x(t),y(t)] or r(t)=[x(t),y(t),z(t)]  */
unitT(r,t):=
block([rp,rpn,T],
   define(rp(t),diff(r,t)),
   define(rpn(t),sqrt(rp(t).rp(t))),
   define(T(t),rp(t)/rpn(t)),
   trigsimp(T(t))
)$
/*the help file for use with help()*/
unitT_help():=block(printf(true,
"unitT computes the unit tangent vector for a vector valued function
 r: R --> R^2 or   r: R --> R^3
of a scalar parameter t   r(t)=[x(t),y(t)] or r(t)=[x(t),y(t),z(t)]
      Example
          unitT(2*[cos(t),sin(t)],t); returns
             [-sin(t),cos(t)]"
)
);

/* unitN computes the unit normal vector for a vector valued function  r: R --> R^3
of a scalar variable r(t)=[x(t),y(t),z(t)]  
  unitN requires unitT */
unitN(r,t):=
block([T,Tp,Tpn,N],
   define(T(t),unitT(r,t)),
   define(Tp(t),diff(T(t),t)),
   define(Tpn(t),sqrt(Tp(t).Tp(t))),
   define(N(t),Tp(t)/Tpn(t)),
   trigsimp(N(t)) 
)$
/*the help file for use with help()*/
unitN_help():=block(printf(true,
"unitN computes the unit normal vector for a vector valued function
 r: R --> R^2 or   r: R --> R^3
of a scalar parameter t   r(t)=[x(t),y(t)] or r(t)=[x(t),y(t),z(t)]
      Example
          unitN(2*[cos(t),sin(t)],t); returns
             [-cos(t),-sin(t)]"
)
);

/* unitB computes the unit normal vector for a vector valued function  r: R --> R^3
of a scalar variable r(t)=[x(t),y(t),z(t)]  
  unitB requires unitT and unitN */
unitB(r,t):=
block([T,N,B],
   define(T(t),unitT(r,t)),
   define(N(t),unitN(r,t)),
   define(B(t),[T(t)[2]*N(t)[3]-T(t)[3]*N(t)[2],
                  T(t)[3]*N(t)[1]-T(t)[1]*N(t)[3],
                  T(t)[1]*N(t)[2]-T(t)[2]*N(t)[1]]),
    trigsimp(B(t)) 
)$
/*the help file for use with help()*/
unitB_help():=block(printf(true,
"unitB computes the unit normal vector for a vector valued function  r: R --> R^3
of a scalar variable r(t)=[x(t),y(t),z(t)]  
      Example
          unitB(2*[cos(t),sin(t),t],t); returns
             [sin(t)/sqrt(2),-cos(t)/sqrt(2),1/sqrt(2)]"
)
);



/*  integratePaths computes the path integral of a real valued integrand 
   f(x,y): R^n --> R,  on path r(t): R --> R^n, with real parameter t from a to b  */
integratePaths(f,r,t,a,b):=block(
[f1,f2,dr,Iout],
f1:subst(x=r[1],f),
f2:subst(y=r[2],f1),
dr:sqrt(diff(r,t).diff(r,t)),
Iout: integrate(trigsimp(f2*dr),t,a,b),
Iout
);
/*the help file for use with help()*/
integratePaths_help():=block(printf(true,
"integratePaths computes the path integral of a real integrand 
   f(x,y): R^2 --> R,  on a path r(t): R --> R^2, with real parameter t from a to b  
      Example
          integratePaths(x*y,[t,1-t],t,0,1); returns
             1/(3*sqrt(2))"
)
);


/* integratePathv2: path integral of a vector integrand F(x,y) on path  r(t) in R^2, t from a to b */
integratePathv2(H,r,t,a,b):=block(
[H1,H2,I],
H2:psubst([x=r[1],y=r[2]],H),
/* H2:subst(y=r[2],H1), */
I: integrate( trigsimp(H2.diff(r,t)),t,a,b),
I
);
/*the help file for use with help()*/
integratePathv2_help():=block(printf(true,
"integratePathv2 computes path integral of a vector integrand 
  F(x,y):R^2 --> R^2 on a path  r(t) in R^2, with real parameter t from a to b  
      Example
          integratePathv2([-y,x^2],[t,1-t],t,0,1); returns
             -5/6"
)
);


/*integratePathv3: path integral of a vector integrand F(x,y,z) on path  r(t) in R^3, t from a to b */
integratePathv3(H,r,t,a,b):=block(
[H1,H2,H3,I],
/*H1:subst(x=r[1],H),
H2:subst(y=r[2],H1),
H3:subst(z=r[3],H2), */
H3:psubst([x=r[1],y=r[2],z=r[3]],H),
I: integrate( trigsimp(H3.diff(r,t)),t,a,b),
I
);
/*the help file for use with help()*/
integratePathv3_help():=block(printf(true,
"integratePathv3 computes path integral of a vector integrand 
  F(x,y,z):R^3 --> R^3 on a path  r(t) in R^3, with real parameter t from a to b  
      Example
          integratePathv3([-y,x^2,x*y*z],[t,1-t,t+2],t,0,1);returns
            -5/12"
)
);



/* Three Maxima functions for the multivariable calculus operators
   grad, div, and curl*/
   
   grad(f,x,y,z):=[diff(f,x),diff(f,y),diff(f,z)]$
   /*the help file for use with help()*/
grad_help():=block(printf(true,
"grad(f,x,y,z) computes the gradient of the scalar-valued function f:R^3-->R  
      Example
          grad(2*x*y+z^2,x,y,z); returns
            [2*y,2*x,2*z]"
)
);
   
   div(f,x,y,z):=diff(f[1],x)+diff(f[2],y)+diff(f[3],z)$
    /*the help file for use with help()*/
div_help():=block(printf(true,
"div(f,x,y,z) computes the divergence of the vector-valued function F:R^3-->R^3  
      Example
          div([x,2*y,3*x*z],x,y,z);returns
            3*x+3"
)
);  
   curl(f,x,y,z):=[ diff(f[3],y)-diff(f[2],z),
                     diff(f[1],z)-diff(f[3],x),
					  diff(f[2],x)-diff(f[1],y) ]$
					      /*the help file for use with help()*/
curl_help():=block(printf(true,
"curl(f,x,y,z) computes the curl of the vector-valued function F:R^3-->R^3  
      Example
          curl([x,2*y,3*x*z],x,y,z);returns
            [0,-3*z,0]"
)
);