91影视

The goal is to demonstrate a conservative vector field whose divergence is uniformly equal to zero in all directions.${\mathrm{鈩�}}^3$

If $\mathbf{F}\left(x_y,y,z\right)$is a conservative vector field, then.

$\mathbf{F}(x,y,z)=\mathbb{V}f(x,y,z)$

The divergence of a vector field $\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is defined as follows $divF=鈭�.F$

Then the divergence of conservative vector field$\mathbf{F}(x,y,z)=鈭�f(x,y,z)$will be,

$divF=鈭�.F$

$$\begin{gathered} =鈭�.鈭�f \\ =(\frac{鈭�}{鈭�x}i+\frac{鈭�}{鈭�y}j+\frac{鈭�}{鈭�z}k).(\frac{鈭�f}{鈭�x}i+\frac{鈭�f}{鈭�y}j+\frac{鈭�f}{鈭�z}k) \\ =\frac{鈭�}{鈭�x}\left(\frac{鈭�f}{鈭�x}\right)+\frac{鈭�}{鈭�y}\left(\frac{鈭�f}{鈭�y}\right)+\frac{鈭�}{鈭�z}\left(\frac{鈭�f}{鈭�z}\right) \\ =\frac{{鈭�}^{2}f}{鈭�x^2}+\frac{{鈭�}^{2}f}{鈭�y^2}+\frac{{鈭�}^{2}f}{鈭�z^2} \end{gathered}$$

Hence, a conservative vector field whose divergence is uniformly equal to zero in ${\mathrm{鈩�}}^3$satisfies the following conditions,

$\frac{{鈭�}^{2}f}{鈭�x^2}=0,\frac{{鈭�}^{2}f}{鈭�y^2}=0,\frac{{鈭�}^{2}f}{鈭�z^2}=0$

Suppose that , $f=x+y+z$, then

$$\begin{gathered} 鈭�f=(\frac{鈭�}{鈭�x}i+\frac{鈭�}{鈭�y}j+\frac{鈭�}{鈭�z}k)(x+y+z) \\ =\frac{鈭�}{鈭�x}(x+y+z)i+\frac{鈭�}{鈭�y}(x+y+z)j+\frac{鈭�}{鈭�z}(x+y+z)k \\ =1i+1j+1k \\ =i+j+k \end{gathered}$$

Then, the vector field $F(x,y,z)=i+j+k$is a conservative vector field, such that $F=鈭�f$

Where $F=x+y+z$

Notice that,

$$\begin{gathered} \frac{{鈭�}^{2}f}{鈭�x^2}=\frac{{鈭�}^2}{鈭�x^2}(x+y+z) \\ =\frac{鈭�}{鈭�x}\left(\frac{鈭�}{鈭�x}\right(x+y+z\left)\right) \\ =\frac{鈭�}{鈭�x}\left(1\right) \\ =0 \\ \frac{{鈭�}^{2}f}{鈭�y^2}==\frac{{鈭�}^2}{鈭�y^2}(x+y+z) \\ =\frac{鈭�}{鈭�y}\left(\frac{鈭�}{鈭�y}\right(x+y+z\left)\right) \\ =\frac{鈭�}{鈭�y}\left(1\right) \\ =0 \\ \frac{{鈭�}^{2}f}{鈭�z^2}=\frac{{鈭�}^2}{鈭�z^2}(x+y+z) \\ =\frac{鈭�}{鈭�z}\left(\frac{鈭�}{鈭�z}\right(x+y+z\left)\right) \\ =\frac{鈭�}{鈭�z}\left(1\right) \\ =0 \end{gathered}$$

Hence, the divergence for this conservative vector field $F(x,y,z)=i+j+k$ is uniformly equal to zero in .

Therefore, an example of a conservative vector field whose divergence is uniformly equal to zero in ${\mathrm{鈩�}}^3$is