> # bellenot\ #73\ u:=(x,y,z)->ln(x+2*y^2+3*z^3);\ diff( u(x,y,z), x, y, z); 2 3 u := (x,y,z) -> ln(x + 2 y + 3 z ) 2 y z 72 ------------------ 2 3 3 (x + 2 y + 3 z ) > #77\ u:=(x,y,z)->1/sqrt(x^2+y^2+z^2);\ ans:=diff(u(x,y,z),x,x)+diff(u(x,y,z),y,y)+diff(u(x,y,z),z,z);\ simplify(ans); 1 u := (x,y,z) -> ------------------ 2 2 2 sqrt(x + y + z ) 2 2 x 3 y ans := 3 ----------------- - ----------------- + 3 ----------------- 2 2 2 5/2 2 2 2 3/2 2 2 2 5/2 (x + y + z ) (x + y + z ) (x + y + z ) 2 z + 3 ----------------- 2 2 2 5/2 (x + y + z ) 0 -------------------------------------------------------------------------------- > #6\ with(linalg):\ grad(E^x*sin(y),[x,y]); x x [ E sin(y), E cos(y) ] -------------------------------------------------------------------------------- > g:=(x,y)->[exp(x)*sin(y), exp(x)*cos(y)];\ g(1,Pi/4);\ dotprod(g(1,Pi/4),[-1/sqrt(5),2/sqrt(5)]); g := (x,y) -> [exp(x) sin(y), exp(x) cos(y)] 1/2 1/2 [1/2 exp(1) 2 , 1/2 exp(1) 2 ] 1/2 1/2 1/10 exp(1) 2 5 -------------------------------------------------------------------------------- > # critical points section\ readlib(unassign): > #6\ unassign('x','y');\ f:=(x,y)->2*x^3+x*y^2+5*x^2+y^2;\ fx:=diff(f(x,y),x);\ fy:=diff(f(x,y),y);\ critical := [solve ( { diff(f(x,y),x)=0, diff(f(x,y),y)=0 } )];\ Delta:= unapply( diff( f(x,y),x,x)*diff( f(x,y),y,y) - (diff(f(x,y),x,y))^2, x,y);\ fxx:=unapply(diff(f(x,y),x,x),x,y);\ for i from 1 to nops(critical)\ do\ unassign ( 'x', 'y' );\ critical[i];\ evalf(critical[i]);\ assign(critical[i] );\ dd:=Delta(x,y);\ f11:=fxx(x,y);\ od; 3 2 2 2 f := (x,y) -> 2 x + x y + 5 x + y 2 2 fx := 6 x + y + 10 x fy := 2 x y + 2 y critical := [{y = 0, x = 0}, {x = -5/3, y = 0}, {x = -1, y = 2}, {x = -1, y = -2}] 2 Delta := (x,y) -> (12 x + 10) (2 x + 2) - 4 y fxx := (x,y) -> 12 x + 10 {y = 0, x = 0} {y = 0, x = 0} dd := 20 f11 := 10 {x = -5/3, y = 0} {y = 0, x = -1.666666667} dd := 40/3 f11 := -10 {x = -1, y = 2} {x = -1., y = 2.} dd := -16 f11 := -2 {x = -1, y = -2} {y = -2., x = -1.} dd := -16 f11 := -2 -------------------------------------------------------------------------------- > #7\ unassign('x','y');\ f:=(x,y)->x^3-3*x*y+y^3;\ fx:=diff(f(x,y),x);\ fy:=diff(f(x,y),y);\ critical := [solve ( { diff(f(x,y),x)=0, diff(f(x,y),y)=0 } )];\ Delta:= unapply( diff( f(x,y),x,x)*diff( f(x,y),y,y) - (diff(f(x,y),x,y))^2, x,y);\ fxx:=unapply(diff(f(x,y),x,x),x,y);\ for i from 1 to nops(critical)\ do\ unassign ( 'x', 'y' );\ critical[i];\ evalf(critical[i]);\ assign(critical[i] );\ dd:=Delta(x,y);\ f11:=fxx(x,y);\ od; 3 3 f := (x,y) -> x - 3 x y + y 2 fx := 3 x - 3 y 2 fy := - 3 x + 3 y critical := [{y = 0, x = 0}, {y = 1, x = 1}, 2 2 {y = RootOf(_Z + _Z + 1), x = - RootOf(_Z + _Z + 1) - 1}] Delta := (x,y) -> 36 x y - 9 fxx := (x,y) -> 6 x {y = 0, x = 0} {y = 0, x = 0} dd := -9 f11 := 0 {y = 1, x = 1} {y = 1., x = 1.} dd := 27 f11 := 6 2 2 {y = RootOf(_Z + _Z + 1), x = - RootOf(_Z + _Z + 1) - 1} {y = - .5000000000 - .8660254038 I, x = - .5000000000 + .8660254038 I} 2 2 dd := 36 (- RootOf(_Z + _Z + 1) - 1) RootOf(_Z + _Z + 1) - 9 2 f11 := - 6 RootOf(_Z + _Z + 1) - 6 -------------------------------------------------------------------------------- > #8\ unassign('x','y');\ f:=(x,y)->y*sqrt(x)-y^2-x+6*y;\ fx:=diff(f(x,y),x);\ fy:=diff(f(x,y),y);\ critical := [solve ( { diff(f(x,y),x)=0, diff(f(x,y),y)=0 } )];\ Delta:= unapply( diff( f(x,y),x,x)*diff( f(x,y),y,y) - (diff(f(x,y),x,y))^2, x,y);\ fxx:=unapply(diff(f(x,y),x,x),x,y);\ for i from 1 to nops(critical)\ do\ unassign ( 'x', 'y' );\ critical[i];\ evalf(critical[i]);\ assign(critical[i] );\ dd:=Delta(x,y);\ f11:=fxx(x,y);\ od; 2 f := (x,y) -> y sqrt(x) - y - x + 6 y y fx := 1/2 ---- - 1 1/2 x 1/2 fy := x - 2 y + 6 critical := [{y = 4, x = 4}] y 1 Delta := (x,y) -> 1/2 ---- - --- 3/2 4 x x y fxx := (x,y) -> - 1/4 ---- 3/2 x {y = 4, x = 4} {y = 4., x = 4.} 1/2 dd := 1/8 4 - 1/16 1/2 f11 := - 1/16 4 -------------------------------------------------------------------------------- > #12\ unassign('x','y');\ f:=(x,y)->x^2+y^2+1/(x^2*y^2);\ fx:=diff(f(x,y),x);\ fy:=diff(f(x,y),y);\ critical := [solve ( { diff(f(x,y),x)=0, diff(f(x,y),y)=0 } )];\ Delta:= unapply( diff( f(x,y),x,x)*diff( f(x,y),y,y) - (diff(f(x,y),x,y))^2, x,y);\ fxx:=unapply(diff(f(x,y),x,x),x,y);\ for i from 1 to nops(critical)\ do\ unassign ( 'x', 'y' );\ critical[i];\ evalf(critical[i]);\ assign(critical[i] );\ dd:=Delta(x,y);\ f11:=fxx(x,y);\ od; 2 2 1 f := (x,y) -> x + y + ----- 2 2 x y 2 fx := 2 x - ----- 3 2 x y 2 fy := 2 y - ----- 2 3 x y critical := [{y = 1, x = 1}, {x = -1, y = 1}, {x = -1, y = -1}, 2 2 {x = 1, y = -1}, {y = RootOf(_Z + _Z + 1), x = RootOf(_Z + _Z + 1)}, 2 2 {y = RootOf(_Z - _Z + 1), x = - RootOf(_Z - _Z + 1)}, 2 2 {y = RootOf(_Z + _Z + 1), x = - RootOf(_Z + _Z + 1)}, 2 2 {y = RootOf(_Z - _Z + 1), x = RootOf(_Z - _Z + 1)}] / 6 \ / 6 \ 16 Delta := (x,y) -> |2 + -----| |2 + -----| - ----- | 4 2| | 2 4| 6 6 \ x y / \ x y / x y 6 fxx := (x,y) -> 2 + ----- 4 2 x y {y = 1, x = 1} {y = 1., x = 1.} dd := 48 f11 := 8 {x = -1, y = 1} {y = 1., x = -1.} dd := 48 f11 := 8 {x = -1, y = -1} {x = -1., y = -1.} dd := 48 f11 := 8 {x = 1, y = -1} {x = 1., y = -1.} dd := 48 f11 := 8 2 2 {y = RootOf(_Z + _Z + 1), x = RootOf(_Z + _Z + 1)} {y = - .5000000000 - .8660254038 I, x = - .5000000000 - .8660254038 I} / 6 \2 16 dd := |2 + ---------------------| - ---------------------- | 2 6| 2 12 \ RootOf(_Z + _Z + 1) / RootOf(_Z + _Z + 1) 6 f11 := 2 + --------------------- 2 6 RootOf(_Z + _Z + 1) 2 2 {y = RootOf(_Z - _Z + 1), x = - RootOf(_Z - _Z + 1)} {x = - .5000000000 + .8660254038 I, y = .5000000000 - .8660254038 I} / 6 \2 16 dd := |2 + ---------------------| - ---------------------- | 2 6| 2 12 \ RootOf(_Z - _Z + 1) / RootOf(_Z - _Z + 1) 6 f11 := 2 + --------------------- 2 6 RootOf(_Z - _Z + 1) 2 2 {y = RootOf(_Z + _Z + 1), x = - RootOf(_Z + _Z + 1)} {y = - .5000000000 - .8660254038 I, x = .5000000000 + .8660254038 I} / 6 \2 16 dd := |2 + ---------------------| - ---------------------- | 2 6| 2 12 \ RootOf(_Z + _Z + 1) / RootOf(_Z + _Z + 1) 6 f11 := 2 + --------------------- 2 6 RootOf(_Z + _Z + 1) 2 2 {y = RootOf(_Z - _Z + 1), x = RootOf(_Z - _Z + 1)} {x = .5000000000 - .8660254038 I, y = .5000000000 - .8660254038 I} / 6 \2 16 dd := |2 + ---------------------| - ---------------------- | 2 6| 2 12 \ RootOf(_Z - _Z + 1) / RootOf(_Z - _Z + 1) 6 f11 := 2 + --------------------- 2 6 RootOf(_Z - _Z + 1) -------------------------------------------------------------------------------- > #1\ unassign('x','y');\ f:=(x,y,lamba)->x^2-y^2-lamba*(x^2+y^2-1);\ fx:=diff(f(x,y,lamba),x);\ fy:=diff(f(x,y,lamba),y);\ flamba:=diff( f(x,y,lamba),lamba);\ critical := [solve ( { fx=0, fy=0, flamba=0 } )]; 2 2 2 2 f := (x,y,lamba) -> x - y - lamba (x + y - 1) fx := 2 x - 2 lamba x fy := - 2 y - 2 lamba y 2 2 flamba := - x - y + 1 critical := [{y = 0, x = 1, lamba = 1}, {x = -1, y = 0, lamba = 1}, {x = 0, y = 1, lamba = -1}, {x = 0, y = -1, lamba = -1}] -------------------------------------------------------------------------------- > g:=(x,y)->x^2-y^2; g(1,0);g(-1,0);g(0,1);g(0,-1); 2 2 g := (x,y) -> x - y 1 1 -1 -1 -------------------------------------------------------------------------------- >