Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x^2 y \mathbf{i}+\frac{1}{3} x^3 \mathbf{j}+x y \mathbf{k}$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z=y^2-x^2$ and the cylinder $x^2+y^2=1$ oriented counterclockwise as viewed from above.

Find the volume of the solid in the first octant bounded by the cylinder $z=9-y^2$ and the plane $x=2$.

Use polar coordinates and L’Hôpital’s Rule to find the limit. lim_(x, y)→(0, 0) (x² + y²)ln (x² + y²)

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geqslant 0$, note that $r \rightarrow 0^{+}$ as $(x, y) \rightarrow(0,0)$.]

$$\lim _{(x, y) \rightarrow(0,0)} \frac{x^3+y^3}{x^2+y^2}$$

Find all second partial derivatives.

$$z=2 x y^3-3 x^2 y$$

Evaluate the double integral $\iint_{D}x\cos y~dA, D$ is bounded by $y = 0, y = x^2, x = 1$.

Evaluate the double integral x cos y dA, D is bounded by y=0,y=x^2, x=1

Use the divergence theorem to evaluate the surface integral $\iint_{S} \boldsymbol{F} \cdot \mathrm{d} S$, where $F=x z i+y z j+z^{2} k$ and S is the closed surface of the hemisphere $x^{2}+y^{2}+z^{2}=4$, z > 0.

Evaluate the surface integral.

$$\iint_S \mathbf{F} \cdot \mathbf{n}\ dS$$

where F(x,y,z) = xi + yj = zk and S is the part of the paraboloid z = 5 - x squared - y squared lying above the plane z = 1; n points upward

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS, where

$$F(x, y, z) = x^2yz i + yz^2j+z^3e^xy k$$

,S is the part of the sphere

$$x^2+y^2+z^2=5$$

that lies above the plane z = 1, and S is oriented upward.

Use the Divergence Theorem to evaluate ∫_s∫ F·N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = xyi + yzj - yzk S: z = √a²-x²-y², z = 0

Use the Divergence Theorem to evaluate ∫_s∫ F·N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = xyi + 4yj + xzk S: x² + y² + z² = 16

Use the Divergence Theorem to evaluate ∫_s∫ F·N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = x²i - 2xyj + xyz²k S: z = √a²-x²-y², z = 0

Use the Divergence Theorem to evaluate ∫_s∫ F·N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = xi + yj + zk S: x² + y² + z² = 9