Surfaces, fluxes and the theorems of Gauss and Stokes

Find the normal vector and the tangent plane to the surface

$$\mathbf{r}(u,v)=(u^2,v,uv)$$

at the point corresponding to $(u_0,v_0)=(1,1)$.

Solution available to those who have unlocked the subject.

Write the equation of the tangent plane to the graph of

$$f(x,y)=x{e}^y$$

at the point with coordinates $(x_0,y_0)=(2,0)$.

Solution available to those who have unlocked the subject.

Compute the area of the portion of the plane

$$z=4-2x-2y$$

contained in the first octant ($x\ge 0$, $y\ge 0$, $z\ge 0$).

Solution available to those who have unlocked the subject.

Compute the area of the portion of the sphere $x^2+y^2+z^2=9$ with $z\ge \frac{3}{2}$.

Solution available to those who have unlocked the subject.

Compute the area of the portion of the paraboloid

$$z=x^2+y^2$$

with $z\le 4$.

Solution available to those who have unlocked the subject.

Compute

$${\iint }_{\Sigma }{z}^{2}d\sigma ,$$

where $\Sigma$ is the sphere $x^2+y^2+z^2=1$.

Solution available to those who have unlocked the subject.

Compute

$${\iint }_{\Sigma }{y}^{2}d\sigma ,$$

where $\Sigma$ is the lateral surface of the cylinder $x^2+y^2=1$ with $0\le z\le 2$.

Solution available to those who have unlocked the subject.

Compute the divergence and the curl of the field

$$\mathbf{F}(x,y,z)=(x^{2}z,xyz,-z^{2}y)$$

at the point $(1,1,1)$.

Solution available to those who have unlocked the subject.

Find $a\in \mathbb{R}$ such that the field

$$\mathbf{F}(x,y,z)=(2xy,a{y}^2,6yz)$$

is solenoidal.

Solution available to those who have unlocked the subject.

Compute the flux of the field

$$\mathbf{F}(x,y,z)=(0,0,x^2+y^2)$$

through the disk $x^2+y^2\le 4$ in the plane $z=3$, oriented upward.

Solution available to those who have unlocked the subject.

Compute the outward flux of the field

$$\mathbf{F}(x,y,z)=(x^3,y^3,z^3)$$

through the sphere $x^2+y^2+z^2=1$.

Solution available to those who have unlocked the subject.

Compute the outward flux of the field

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

through the cube $[0,1]\times [0,1]\times [0,1]$.

Solution available to those who have unlocked the subject.

Compute with Stokes’ theorem the circulation of the field

$$\mathbf{F}(x,y,z)=(2y,3x,z^2)$$

along the circle $x^2+y^2=9$ in the plane $z=0$, traversed counterclockwise as seen from above.

Solution available to those who have unlocked the subject.