SF1626 - Övning 11 - Code

Göran Andersson goeran@kth.se, 110505

9.2

a)

ClearAll["`*"]

𝔽[{x_,y_}]:={y,-1};
𝕣[t_]:={t,1-2t}

(∫_0^1)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

b)

ClearAll["`*"]

𝔽[{x_,y_}]:={y,-1};
𝕒[t_]:={t,1};
𝕓[t_]:={1,1-t}

(∫_0^1)𝔽[𝕒[t]].𝕒'[t]\[DifferentialD]t+(∫_0^2)𝔽[𝕓[t]].𝕓'[t]\[DifferentialD]t

9.3

ClearAll["`*"]

𝔽[{x_,y_}]:={(x^2)+x y,(y^2)-x y};

a)

𝕣[t_]={0,0}+t ({2,2}-{0,0})

(∫_0^1)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

b)

𝕣[t_]:={t,(1/2)(t^2)}

(∫_0^2)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

c)

𝕣1[t_]:={0,0}+t ({2,0}-{0,0});
𝕣2[t_]:={2,0}+t ({2,2}-{2,0});

(∫_0^1)𝔽[𝕣1[t]].(𝕣1)'[t]\[DifferentialD]t+(∫_0^1)𝔽[𝕣2[t]].(𝕣2)'[t]\[DifferentialD]t

9.4

ClearAll["`*"]

𝔽[{x_,y_}]:={y Log[((x^2)/y)],-(x/y)};
𝕣[t_]:={t,(t^2)}

(∫_1^2)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.5

ClearAll["`*"]

𝔽={1,-2};
Δ𝕣={2,2}-{0,1}

A=𝔽.Δ𝕣

9.9

Med Greens formel

ClearAll["`*"]

𝒟=(x^2)+(y^2)≤4;
{P,Q}={(x^3)-(x^2)y,x (y^2)};

Integrate[((∂_x)Q-(∂_y)P) Boole[𝒟],{x,-∞,∞},{y,-∞,∞}]

Direkt beräkning

ClearAll["`*"]

𝕣[t_]:={2 Cos[t],2 Sin[t]};
𝕣'[t]

𝔽[{x_,y_}]:={(x^3)-(x^2)y,x (y^2)}

(∫_0^(2 π))𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.10

Med Greens formel

ClearAll["`*"]

𝒟=(((x-a))^2)+(((y-b))^2)≤(r^2);
{P,Q}={(y^2),(x^2)};

Integrate[((∂_x)Q-(∂_y)P) Boole[𝒟],{x,-∞,∞},{y,-∞,∞},Assumptions->{{a,b,r}∈Reals,r>0}]

Direkt beräkning

ClearAll["`*"]

𝕣[t_]:={a+r Cos[t],b+r Sin[t]};
𝕣'[t]

𝔽[{x_,y_}]:={(y^2),(x^2)}

(∫_0^(2 π))𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.11

Med Greens formel

(∫_Γ)P dx+Q dy=(∫_Δ)P dx+Q (Underscript[dy, (Underscript[⏟, 0])])-(Underscript[∫∫, 𝒟]) (((∂Q)/(∂x))-((∂P)/(∂y)))dx dy

ClearAll["`*"]

𝒟=(x^2)+(y^2)≤1&&y≥0;
{P[x_,y_],Q[x_,y_]}={(x^2)-y+2 Log[1+y],((((1+x))^2)/(1+y))};

(∫_(-1)^1)P[x,0]\[DifferentialD]x-(∫_(-∞)^∞)(∫_(-∞)^∞)((∂_x)Q[x,y]-(∂_y)P[x,y]) Boole[𝒟]\[DifferentialD]y\[DifferentialD]x

Direkt beräkning

ClearAll["`*"]

𝕣[t_]:={ Cos[t], Sin[t]};
𝕣'[t]

𝔽[{x_,y_}]:={(x^2)-y+2 Log[1+y],((((1+x))^2)/(1+y))}

(∫_π^0)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.14

ClearAll["`*"]

𝕣[t_]:={Cos[t],2Sin[t]};
𝔽[{x_,y_}]:={(y^3),(x^3)};

A=(∫_(π/2)^(2π))𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.23

a)

ClearAll["`*"]

𝕣[t_]:={t (1-t),(t^2) (1-t)}

ParametricPlot[𝕣[t],{t,0,1},
Epilog->{Arrow[{𝕣[0.1],𝕣[0.11]}]},
AxesLabel->{"x","y"}
]

b)

𝔽[{x_,y_}]:={0,x}

A=(∫_0^1)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

9.24

ClearAll["`*"]

𝕣[t_]:={t -Sin[t],1-Cos[t]};
𝕩[t_]:={t,0}

ParametricPlot[𝕣[t],{t,2π,0},
Epilog->{Arrow[{𝕣[5],𝕣[4.8]}],Arrow[{𝕩[1.5],𝕩[1.7]}]},
AxesLabel->{"x","y"}
]

𝔽[{x_,y_}]:={0,x}

A=(∫_0^(2π))𝔽[𝕩[t]].𝕩'[t]\[DifferentialD]t+(∫_(2π)^0)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

10.2

a)

ClearAll["`*"];

𝕣[t_]:={0,0,0}+t {1,1,1};

𝔽[{x_,y_,z_}]:={3(x^2)+6y,-14y z,20x (z^2)}

(∫_0^1)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t

b)

ClearAll["`*"];
𝕠={0,0,0};𝕩={1,0,0};𝕪={0,1,0};𝕫={0,0,1};

𝕣1[t_]:=𝕠+t 𝕩;
𝕣2[t_]:=𝕣1[1]+t 𝕪;
𝕣3[t_]:=𝕣2[1]+t 𝕫

𝔽[{x_,y_,z_}]:={3(x^2)+6y,-14y z,20x (z^2)}

(∫_0^1)𝔽[𝕣1[t]].𝕣1'[t]\[DifferentialD]t+(∫_0^1)𝔽[𝕣2[t]].𝕣2'[t]\[DifferentialD]t+(∫_0^1)𝔽[𝕣3[t]].𝕣3'[t]\[DifferentialD]t

c)

ClearAll["`*"];

𝕣[t_]:={t,(t^2),(t^3)};

𝔽[{x_,y_,z_}]:={3(x^2)+6y,-14y z,20x (z^2)}

(∫_0^1)𝔽[𝕣[t]].𝕣'[t]\[DifferentialD]t