91影视

Take a look at the vector field below.:

$\mathbf{F}(x,y,z)=(1鈭�yz\sin 鈦�(xyz\left)\right)\mathbf{i}鈭�(1+xz\sin 鈦�(xyz\left)\right)\mathbf{j}+(1鈭�xy\sin 鈦�(xyz\left)\right)\mathbf{k}$

The goal is to assess the integral.${鈭�}_{S}curl\mathbf{F}(x,y,z)路\mathbf{n}dS$, where the surface Siis defined as:

The surface S is the hyperbolic paraboloid's portion.$z=x^2-y^2$that lies inside the cylinder $4x^2+9y^2=36$with upwards-pointing normal vector.

$\mathbf{F}(x,y,z)=(1鈭�yz\sin 鈦�(xyz\left)\right)\mathbf{i}鈭�(1+xz\sin 鈦�(xyz\left)\right)\mathbf{j}+(1鈭�xy\sin 鈦�(xyz\left)\right)\mathbf{k}$

A vector field's curl$\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is defined as follows:

$\mathrm{curl}鈦�\mathbf{F}(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\mathrm{鈭�}}{\mathrm{鈭�}x}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}y}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\\ F_1(x,y,z)& F_2(x,y,z)& F_3(x,y,z)\end{array}\right|$

$=\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}y}鈭�\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}z}\right]\mathbf{i}鈭�\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}z}\right]\mathbf{j}+\left[\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right]\mathbf{k}$

]]The vector field's curl is then applied. $\mathbf{F}(x,y,z)$will be,

$\mathrm{curl}鈦�\mathbf{F}(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\mathrm{鈭�}}{\mathrm{鈭�}x}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}y}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\\ (1鈭�yz\sin 鈦�(xyz\left)\right)& 鈭�(1+xz\sin 鈦�(xyz\left)\right)& (1鈭�xy\sin 鈦�(xyz\left)\right)\end{array}\right|$

$=\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(1鈭�xy\sin 鈦�(xyz\left)\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}(鈭�(1+xz\sin 鈦�\left(xyz\right)\left)\right)\right]\mathbf{i}$

$鈭�\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}(1鈭�xy\sin 鈦�(xyz\left)\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}(1鈭�yz\sin 鈦�(xyz\left)\right)\right]\mathbf{j}$

$+\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}(鈭�(1+xz\sin 鈦�\left(xyz\right)\left)\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(1鈭�yz\sin 鈦�(xyz\left)\right)\right]\mathbf{k}$

$$\begin{gathered} 0i+oj+ok \\ =0 \end{gathered}$$

Then, the value of curl$\mathbf{F}(x,y,z)路\mathbf{n}$will be,

$$\begin{gathered} 鈦�\mathbf{F}(x,y,z)鈰�\mathbf{n}=\mathbf{0}鈰�\mathbf{n} \\ \mathbf{=}\mathbf{0} \end{gathered}$$

As a result, the required integral is

$$\begin{gathered} {鈭�}_S鈥�\mathrm{curl}鈦�\mathbf{F}(x,y,z)鈰�\mathbf{n}dS={鈭�}_S鈥�0dS \\ =0 \end{gathered}$$

As a result, the required integral is${鈭�}_{S}curl\mathbf{F}(x,y,z)路\mathbf{n}dS=0$.