(*
	Advanced Calculus Package
	
	Dennis M. Snow
	Dept. Mathematics
	Notre Dame
	October 1991
	
	Fixed Jacobian, J[], for three variables, January 1996
	Removed Cross[], since it is now built in, January 1997
*)


BeginPackage["Calculus`Advanced`"]

(* Declarations *)

	Global`Vars::usage =
		"Vars is a list of variables used by default in calculating
		Advanced Calculus functions. Initially Vars = {x,y,z}."

	Norm::usage =
		"Norm[v] gives the norm (or length) of a vector v."

	Pr::usage =
		"Pr[v, w] gives the projection of a vector v onto a vector w."

(*	Cross::usage =
		"Cross[v, w] gives the cross product of the 3-dimensional vectors v 
		and w." *)

	Grad::usage =
		"Grad[f, v] gives the gradient of f. The optional argument v is a
		list of the independent variables."

	CriticalPoints::usage =
		"CriticalPoints[f, v] finds the critical points of f. The optional 
		argument v is a list of the independent variables."

	Discriminant::usage =
		"Discriminant [f,v] gives the discriminant of a function f. The 
		optional argument v is a list of the independent variables."

	Lagrange::usage =
		"Lagrange[f, g==c, v] finds the critical points of f subject to the 
		constraint g == c. The optional argument v is a list of the 
		independent variables."

	Curl::usage =
		"Curl[F, v] gives the curl of the 3-dimensional vector function F. 
		The optional argument v is a list of the independent variables."

	Div::usage =
		"Div[F, v] gives the divergence of the vector function F.
		 The optional argument v is a list of the independent
		 variables."

	Vector::usage =
		"Vector[{x0,y0,z0},{x1,y1,z1}] creates a thick line segment from 
		{x0,y0,z0} to {x1,y1,z1} as a Graphics3D object."

	VectorField::usage =
		"VectorField[F, {x, x0, x1}, {y, y0, y1}] plots the
		2-dimensional vector field F on a grid of points {x,y}
		where x0 <= x <= x1 and y0 <= y <= y1.
		VectorField[F, {x, x0, x1, dx}, {y, y0, y1, dy}] will
		space the grid points {dx,dy} apart.
		Options are passed to Show[]."

	J::usage =
		"J[f,g,{u,v}] gives the Jacobian determinant of the coordinate
		transformation x = f(u,v), y = g(u,v). J[f,g,h,{u,v,w}] gives
		the Jacobian determinant of the coordinate transformation
		x = f(u,v,w), y = g(u,v,w), z = h(u,v,w)."

	LineIntegrate::usage =
		"LineIntegrate[f, r, {t, a, b}, v] computes the line integral
		 of the function f along the curve parameterized by the vector
		 function r from t = a to t = b. The optional argument v is a
		 list of the independent variables."

	FlowIntegrate::usage =
		"FlowIntegrate[F, r, {t, a, b}, v] computes the flow integral
		 of the vector field F along the curve parameterized by the
		 vector function r from t = a to t = b. The optional argument v
		 is a list of the independent variables."
 
(* Definitions *)

Begin["`Private`"]

	Clear[Global`x,Global`y,Global`z]

	Global`Vars = {Global`x,Global`y,Global`z}

	Norm[v_?VectorQ] := Sqrt[v.v]

	Pr[v_?VectorQ,w_?VectorQ] := (v.w) / (w.w) w

(*	Cross[v_?VectorQ, w_?VectorQ] :=
		{v[[2]]w[[3]]-v[[3]]w[[2]],v[[3]]w[[1]]-v[[1]]w[[3]],
		v[[1]]w[[2]]-v[[2]]w[[1]]} *)

	Grad[f_, v_?VectorQ] := D[f,#]&/@v

	Grad[f_] := Grad[f,Global`Vars]

	Discriminant[f_,{x_, y_}] := Simplify[D[f,x,x] D[f,y,y] - D[f,x,y]^2]

	Discriminant[f_] := Discriminant[f,{Global`Vars[[1]],Global`Vars[[2]]}]

	CriticalPoints[f_,v_?VectorQ]:= Solve[Grad[f,v] == 0]

	CriticalPoints[f_] := CriticalPoints[f,Global`Vars]

	Lagrange[f_,g_==c_?NumberQ,v_?VectorQ] :=
		Module[{lambda},Solve[Eliminate[{Grad[f,v]==lambda Grad[g,v],g == c},
lambda]]]

	Lagrange[f_,g_==c_?NumberQ] := Lagrange[f,g==c,Global`Vars]

	Curl[F_?VectorQ,{x_,y_,z_}] := {D[F[[3]],y] - D[F[[2]],z],
		D[F[[1]],z] - D[F[[3]],x], D[F[[2]],x] - D[F[[1]],y]}

	Curl[F_?VectorQ] := Curl[F,Global`Vars]

	Div[F_?VectorQ,v_?VectorQ] := Sum[D[F[[i]],v[[i]]], {i,Length[v]}]
	
	Div[F_?VectorQ] := Div[F,Global`Vars]

	Vector[a_List, b_List] :=
		Graphics3D[{Thickness[0.01], Line[{a,b}]}]

	VectorField[F_?VectorQ, {x_, x0_, x1_, dx___}, {y_, y0_, y1_, dy___}, 
Opts___]:=
		Show[Graphics[{PointSize[0.015], Table[{Point[{x,y}], Line[{{x,y},{x,y}+
F}]},
		{x, x0, x1, dx}, {y, y0, y1, dy}]} ], Axes->Automatic, AspectRatio->
Automatic, Opts]

	J[x_,y_,{u_,v_}] := Det[{{D[x,u], D[x,v]}, {D[y,u], D[y,v]}}]

	J[x_,y_,z_,{u_,v_,w_}] := Det[{{D[x,u], D[x,v], D[x,w]},
		{D[y,u], D[y,v], D[y,w]},{D[z,u], D[z,v], D[z,w]}}]
	
	LineIntegrate[ f_, r_?VectorQ, {t_, a_, b_}, v_ ] :=
		Integrate[(f/.Inner[Rule,Take[v,Length[r]],r,List])*Norm[D[r,t]], {t,a,b} ]

	LineIntegrate[ f_, r_, {t_, a_, b_} ] :=
		LineIntegrate[f,r,{t,a,b},Global`Vars]

	FlowIntegrate[ F_, r_?VectorQ, {t_, a_, b_}, v_ ] :=
		Integrate[(F/.Inner[Rule,Take[v,Length[r]],r,List]).D[r,t],
			{t,a,b} ] /; Length[F] == Length[r]	

	FlowIntegrate[ F_, r_?VectorQ, {t_, a_, b_}] :=
		FlowIntegrate[ F, r, {t, a, b},Global`Vars] 	

End[]	

EndPackage[]