Section 16.3: Fundamental Theorem of Line Integrals
Homework
MAT329
Andy Long

#4, p. 1106: yep!

F[x_,y_]:={E^x Sin[y],E^x Cos[y]}
D[F[x,y],x]
D[F[x,y],y]
f[x_,y_]:=E^x Sin[y]
D[f[x,y],{{x,y}}]

#6, p. 1106: nope!

F[x_,y_]:={3x^2-2y^2,4x y+3}
D[F[x,y],x]
D[F[x,y],y]

#20, p. 1107: yep!

F[x_,y_]:={Sin[y],x Cos[y]-Sin[y]}
f[x_,y_]:=x Sin[y]+Cos[y]
D[F[x,y],x]
D[F[x,y],y]
f[1,Pi]-f[2,0]

#28, p. 1107:

f[x_,y_]:= Sin[x-2y]
F[x_,y_]=D[f[x,y],{{x,y}}]
VectorPlot[F[x,y],{x,-Pi,Pi},{y,-Pi,Pi}]

Note: F is a gradient field.

f[x+2Pi,y]-f[x,y]
f[Pi/2,0]-f[0,0]