91影视

The given equation is$\mathrm{V}=2\mathrm{xyi}+({\mathrm{x}}^2鈭�2\mathrm{x})\mathrm{j}鈭�{\mathrm{x}}^2{\mathrm{z}}^2\mathrm{k}$

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

Calculate the expression $鈭�(\text{肠耻谤濒痴)脳}\mathrm{苍诲蟽}$for$\mathrm{V}=2\mathrm{xyi}+({\mathrm{x}}^2鈭�2\mathrm{x})\mathrm{j}鈭�{\mathrm{x}}^2{\mathrm{z}}^2\mathrm{k}$ over the surface $\mathrm{z}=9鈭�{\mathrm{x}}^2鈭�9{\mathrm{y}}^2$above the $\mathrm{x}-\mathrm{y}$plane(i.e., for $\mathrm{z}鈮�0$).This surface is bounded by the ellipse$9鈭�{\mathrm{x}}^2鈭�9{\mathrm{y}}^2=0$ which means that, using Stokes's theorem, the original integral can be calculated over any surface enclosed by this ellipse, which is taken to be the surface on the $\mathrm{x}-\mathrm{y}$plane (i.e., the ellipse itself). The $\mathrm{curl}\mathrm{V}$of is derived as shown below.

$\mathrm{curlV}=\left(\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{z}}}{\mathrm{鈭�}\mathrm{y}}鈭�\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{y}}}{\mathrm{鈭�}\mathrm{z}}\right)\mathrm{i}+\left(\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{x}}}{\mathrm{鈭�}\mathrm{z}}鈭�\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{z}}}{\mathrm{鈭�}\mathrm{x}}\right)\mathrm{j}+\left(\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{y}}}{\mathrm{鈭�}\mathrm{x}}鈭�\frac{\mathrm{鈭�}{\mathrm{V}}_{\mathrm{x}}}{\mathrm{鈭�}\mathrm{y}}\right)\mathrm{k}$

$=(0鈭�0)\mathrm{i}+\left(0鈭�2{\mathrm{xz}}^2\right)\mathrm{j}+(2\mathrm{x}鈭�2鈭�2\mathrm{x})\mathrm{k}$

$=-2\mathrm{K}$

Therefore, the integral of $-2\mathrm{K}$over the area of an ellipse is $\frac{{\mathrm{x}}^2}{9}+{\mathrm{y}}^2=1$.

Its normal vector is and its area is which gives the solution shown below.

$鈭�\left(\text{curlV}\right)脳\mathrm{苍诲蟽}=鈭�(鈭�2)\mathrm{k}脳\mathrm{办诲蟽}$

$=-2\left(3\mathrm{蟺}\right)$

$=-6\mathrm{蟺}$

Hence, the solution is $鈭�\left(\text{curlV}\right)脳\text{nd}\mathrm{蟽}=鈭�6\mathrm{蟺}$.