91影视

The given is the function $鈭�{鈭�}_{s}f(x,y,z)路nds=0$

The objective is to show that the function does not follow that $F$ is conservative

The goal is to demonstrate that if,

${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS=0$

It is not the case $\mathbf{F}$is conservative.

A vector field If and only if the following conditions are met, it is conservative.

Consider the vector field below:

$\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$

This vector field is for

$F_1(x,y,z)=y+z,F_2(x,y,z)=2x+xz,and{F}_3(x,y,z)=2xy+y$

Take note of this:

role="math" $$\begin{gathered} \frac{鈭�F_3}{鈭�y}=\frac{鈭�}{鈭�y}(2xy+y) \\ =2x+1 \end{gathered}$$

role="math" $$\begin{gathered} \frac{鈭�F_2}{鈭�z}=\frac{鈭�}{鈭�z}(2x+xz) \\ =x \end{gathered}$$

As a result, in this example,

$\frac{鈭�F_3}{鈭�y}鈮�\frac{鈭�F_2}{鈭�z}$

Consequently, this vector field $\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$is a vector field that is not conservative.

It is sufficient to establish that the integral is necessary ${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS=0$for the vector field that isn't conservative$\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$.

Proof:

To establish the above, use the Divergence Theorem.

According to the Divergence Theorem,

"Let $W$be a bounded region in ${\mathrm{鈩�}}^3$with a smooth or piecewise-smooth closed oriented surface as its boundary $S$. If a vector field exists,

$\mathbf{F}(x,y,z)$is defined on an open region containing $W$, then,

${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS={鈭�}_{W}div\mathbf{F}(x,y,z)dV$

Here $\mathbf{n}$is the normal vector pointing outwards. "

First, determine the vector field's divergence $\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$.

A vector field's divergence is defined as follows:

$=\frac{鈭�F_1}{鈭�x}+\frac{鈭�F_2}{鈭�y}+\frac{鈭�F_3}{鈭�z}.$

Then there's the vector field's divergence $\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$will be,

Then there's the vector field's divergence $F(x,y,z)=<y+z,2x+xz,2xy+y>willbe,divF=鈭�路F$

$$\begin{gathered} =\frac{鈭�}{鈭�x}(y+z)+\frac{鈭�}{鈭�y}(2x+xz)+\frac{鈭�}{鈭�z}(2xy+y) \\ =0+0+0 \\ =0 \end{gathered}$$

Now evaluate the integral using the Divergence Theorem (1)${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS$as follows:

${鈭�}_S\mathbf{F}路\mathbf{n}dS={鈭�}_{R}div\mathbf{F}dV$

$$\begin{gathered} ={鈭�}_{R}0dV \\ =0 \end{gathered}$$

As a result, in the case of non-conservative vector fields, $\mathbf{F}(x,y,z)=鉄�y+z,2x+xz,2xy+y鉄�$,

${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS=0$

As a result, it has been established that, if ${鈭�}_S\mathbf{F}(x,y,z)路\mathbf{n}dS=0$, it does not follow that $\mathrm{F}$ is conservative.