Vector fields and Green’s theorem

Determine whether the field

$$F(x,y)=(3x^2+2y,2x-5y^4)$$

is conservative on ${\mathbb{R}}^2$ and, if so, determine its potentials.

Solution available to those who have unlocked the subject.

Determine whether the field

$$F(x,y)=(x^{2}y,x{y}^2)$$

is conservative on ${\mathbb{R}}^2$.

Solution available to those who have unlocked the subject.

Determine whether the field

$$F(x,y)=(2x{e}^y,x^2{e}^y-2y)$$

is conservative on ${\mathbb{R}}^2$ and, if so, determine its potentials.

Solution available to those who have unlocked the subject.

Compute the work of the field

$$F(x,y)=(y^2,2xy)$$

along the curve $r(t)=\left(t^3,\sin \frac{\pi t}{2}\right)$, $t\in [0,1]$.

Solution available to those who have unlocked the subject.

Compute the work of the field

$$F(x,y)=(y,-x)$$

along the segment from $(1,0)$ to $(0,1)$.

Solution available to those who have unlocked the subject.

Compute the circulation of the field

$$F(x,y)=(xy,x)$$

along the circle $x^2+y^2=1$ traversed counterclockwise.

Solution available to those who have unlocked the subject.

Compute the work of the field

$$F(x,y)=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$$

along any piecewise regular curve contained in the half-plane $x>0$, from $(1,0)$ to $(1,1)$.

Solution available to those who have unlocked the subject.

Compute the circulation of the field

$$F(x,y)=(x-y,x+y)$$

along the circle $x^2+y^2=9$ traversed counterclockwise.

Solution available to those who have unlocked the subject.

Compute the circulation of the field

$$F(x,y)=(x^2,xy)$$

along the boundary of the triangle with vertices $(0,0)$, $(2,0)$, $(0,2)$, traversed counterclockwise.

Solution available to those who have unlocked the subject.

Compute with Green’s formula the area of the region enclosed by the ellipse

$$r(t)=(5\cos t,2\sin t),t\in [0,2\pi ].$$

Solution available to those who have unlocked the subject.

Compute with Green’s formula the area of the quadrilateral with vertices $(0,0)$, $(4,1)$, $(3,4)$, $(1,3)$, listed counterclockwise.

Solution available to those who have unlocked the subject.

Determine whether the field

$$F(x,y,z)=(2xz,3y^2{z}^2,x^2+2y^{3}z)$$

is conservative on ${\mathbb{R}}^3$ and, if so, determine its potentials.

Solution available to those who have unlocked the subject.

Compute the work of the field

$$F(x,y,z)=(yz,xz,xy)$$

along the curve $r(t)=(1+t,1+t^2,1+t^3)$, $t\in [0,1]$.

Solution available to those who have unlocked the subject.

Determine whether the field

$$F(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)$$

is conservative on $A={\mathbb{R}}^2\setminus \{(0,0)\}$ and, if so, determine a potential.

Solution available to those who have unlocked the subject.