Tietokoneharjoitus 5
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Teht. 1
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Input :=
f=Integrate[(10-x u-y v^2)^2,{u,0,1},{v,0,1}]
Output =
2 2
x x (-30 + y) 1500 - 100 y + 3 y
-- + ----------- + -------------------
3 3 15
Input :=
jacobi={D[f,x],D[f,y]}
Output =
2 x -30 + y x -100 + 6 y
{--- + -------, - + ----------}
3 3 3 15
Input :=
Solve[jacobi=={0,0},{x,y}]
Output =
80 50
{{x -> --, y -> --}}
7 7
Input :=
f /. %5
Output =
2 2
x x (-30 + y) 1500 - 100 y + 3 y
{-- + ----------- + -------------------}
3 3 15
Input :=
hesse={{D[f,x,x],D[f,x,y]},{D[f,y,x],D[f,y,y]}}
Output =
2 1 1 2
{{-, -}, {-, -}}
3 3 3 5
Input :=
Det[hesse]
Output =
7
--
45
Input :=
Plot3D[f,{x,5,15},{y,5,10}]
Output =
-SurfaceGraphics-
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Teht. 2
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Input :=
Integrate[1/z,{x,0,Log[4]},{y,0,Log[4]-x},{z,E^(x+y),4}]
Integrate::gener: Unable to check convergence.
Output =
3
Log[4]
-------
6
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Teht. 3
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Input :=
<<LinearAlgebra`CrossProduct`
Input :=
u={x y z,0,y^2}
Output =
2
{x y z, 0, y }
Input :=
r={x,y,z}={t,t,t^2}
Output =
2
{t, t, t }
Input :=
Integrate[Cross[u,D[r,t]],{t,0,2}]
Output =
8 56 32
{-(-), -(--), --}
3 3 5
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Teht. 4
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Input :=
Clear[x,y,z,u,v,w]
Input :=
ratkaisu=Solve[{x==u y^2,y==v z^2,z==w x^2},{x,y,z}]
Output =
{{x -> 0, y -> 0, z -> 0},
1 1
{x -> --------------, y -> --------------,
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
1
z -> --------------},
2/7 4/7 1/7
u v w
1/7 4/7
(-1) (-1)
{x -> -(--------------), y -> --------------,
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
2/7
(-1)
z -> --------------},
2/7 4/7 1/7
u v w
2/7 1/7
(-1) (-1)
{x -> --------------, y -> -(--------------),
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
4/7
(-1)
z -> --------------},
2/7 4/7 1/7
u v w
3/7 5/7
(-1) (-1)
{x -> -(--------------), y -> -(--------------),
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
6/7
(-1)
z -> --------------},
2/7 4/7 1/7
u v w
4/7 2/7
(-1) (-1)
{x -> --------------, y -> --------------,
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
1/7
(-1)
z -> -(--------------)},
2/7 4/7 1/7
u v w
5/7 6/7
(-1) (-1)
{x -> -(--------------), y -> --------------,
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
3/7
(-1)
z -> -(--------------)},
2/7 4/7 1/7
u v w
6/7 3/7
(-1) (-1)
{x -> --------------, y -> -(--------------),
1/7 2/7 4/7 4/7 1/7 2/7
u v w u v w
5/7
(-1)
z -> -(--------------)}}
2/7 4/7 1/7
u v w
Input :=
jacobi=Outer[D,{x,y,z}/.ratkaisu[[2]],{u,v,w}]
Output =
-1 -2 -4
{{----------------, ----------------, -----------------},
8/7 2/7 4/7 1/7 9/7 4/7 1/7 2/7 11/7
7 u v w 7 u v w 7 u v w
-4 -1 -2
{-----------------, ----------------, ----------------},
11/7 1/7 2/7 4/7 8/7 2/7 4/7 1/7 9/7
7 u v w 7 u v w 7 u v w
-2 -4 -1
{----------------, -----------------, ----------------}}
9/7 4/7 1/7 2/7 11/7 1/7 2/7 4/7 8/7
7 u v w 7 u v w 7 u v w
Input :=
Det[jacobi]
Output =
-1
----------
2 2 2
7 u v w
Input :=
rajat={z^2==y,z^2==2y,x^2==z,x^2==2z,y^2==x,y^2==2x}
Output =
2 2 2 2 2 2
{z == y, z == 2 y, x == z, x == 2 z, y == x, y == 2 x}
Input :=
For[i=1,i<=6,i++,Print[
Solve[{rajat[[i]],x==u y^2,y==v z^2,z==w x^2},{u,v,w}]]]
x z
{{u -> --, v -> 1, w -> --}}
4 2
z x
4 x 1 z
{{u -> ---, v -> -, w -> --}}
4 2 2
z x
x y
{{u -> --, v -> --, w -> 1}}
2 4
y x
x 4 y 1
{{u -> --, v -> ---, w -> -}}
2 4 2
y x
y z
{{u -> 1, v -> --, w -> --}}
2 4
z y
1 y 4 z
{{u -> -, v -> --, w -> ---}}
2 2 4
z y
Input :=
Integrate[-Det[jacobi],{u,1/2,1},{v,1/2,1},{w,1/2,1}]
Output =
1
-
7
Input :=