91影视

Let $f(x,y,z)$be a function of three variables defined on a open set containing

$\left(x_0,y_0,z_0\right)\text{and let}\mathbf{u}=鉄�伪,尾,纬鉄�$the point containing a unit vector

The direction derivative in $u$direction is given by

$D_{\mathrm{u}}f\left(x_0,y_0,z_0\right)$

$D_{\mathrm{u}}f\left(x_0,y_0,z_0\right)=\underset{h鈫�0}{\lim }\frac{f\left(x_0+伪h,y_0+尾h,z_0+纬h\right)-f\left(x_0,y_0,z_0\right)}{h}$

limit also exists

The gradient of function is given by

$鈭�f\left(x_0,y_0,z_0\right)=f_x\left(x_0,y_0,z_0\right)\mathrm{i}+f_y\left(x_0,y_0,z_0\right)\mathrm{j}+f_z\left(x_0,y_0,z_0\right)\mathrm{k}$

As $伪,尾,纬,x_0,y_0\text{and}{z}_0$are constants

$F\left(h\right)=f\left(x_0+伪h,y_0+尾h,z_0+纬h\right)$is a single variable function

$鈬�F\left(0\right)=f\left(x_0+伪路0,y_0+尾路0,z_0+纬路0\right)=f\left(x_0,y_0,z_0\right)$

From above equation $D_{\mathbf{u}}f\left(x_0,y_0,z_0\right)=\underset{h鈫�0}{\lim }\frac{F\left(h\right)-F\left(0\right)}{h}=F^{\text{'}}\left(0\right)$

If $x=x_0+伪h,y=y_0+尾h\text{and}z=z_0+纬h$

By chain rule

$F^{\text{'}}\left(h\right)=\frac{鈭�f}{鈭�x}\frac{鈭�x}{鈭�h}+\frac{鈭�f}{鈭�y}\frac{鈭�y}{鈭�h}+\frac{鈭�f}{鈭�z}\frac{鈭�z}{鈭�h}$

$=f_x(x,y,z)\frac{鈭�}{鈭�h}\left(x_0+伪h\right)+f_y(x,y,z)\frac{鈭�}{鈭�h}\left(y_0+尾h\right)+f_z(x,y,z)\frac{鈭�}{鈭�h}\left(z_0+纬h\right)$

$=f_x(x,y,z)(0+伪)+f_y(x,y,z)(0+尾)+f_z(x,y,z)(0+纬)$

$=伪f_x(x,y,z)+尾f_y(x,y,z)+纬f_z(x,y,z)$

If $h=0\text{then}x=x_0,y=y_0\text{and}z=z_0\text{.}$

$F^{\text{'}}\left(0\right)=伪f_x\left(x_0,y_0,z_0\right)+尾f_y\left(x_0,y_0,z_0\right)+纬f_z\left(x_0,y_0,z_0\right)$

$D_{\mathrm{u}}f\left(x_0,y_0,z_0\right)=伪f_x\left(x_0,y_0,z_0\right)+尾f_y\left(x_0,y_0,z_0\right)+纬f_z\left(x_0,y_0,z_0\right)$

$=鈭�f\left(x_0,y_0,z_0\right)路\mathbf{u}$

Hence proved