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Thread: Directional derivative definition via difference quotients

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    Directional derivative definition via difference quotients

    Directional derivative, definition via difference quotients -
    I thought to demostrate in this way :

    $\displaystyle w={f}(x_o,y_o) $

    $\displaystyle \frac{dw}{ds}=\frac{d{f}(x_o,y_o)}{ds} = \lim_{h_x \rightarrow 0, h_y \rightarrow 0}{\frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{\sqrt{h_x^2+h_y^2}}} $

    or

    $\displaystyle \frac{d{f}(x_o,y_o)}{ds} = \lim_{h \rightarrow 0}{\frac{f(x_o + h*cos(\phi), y_o+h*sin(\phi))) - f(x_o,y_o)}{h}} $

    -----------------

    if It increase before $\displaystyle \Delta x = h_x $ , from $\displaystyle P_o \rightarrow P_x$ then $\displaystyle \Delta w_a=f{(x_o+h_x,y_o)}-f{(x_o,y_o)} $

    and after increase $\displaystyle \Delta y = h_y $ from $\displaystyle P_x \rightarrow P_1$ then $\displaystyle \Delta w_b=f{(x_1,y_o+h_y)}-f{(x_1,y_o)} $

    in every case this is equal to $\displaystyle P_o \rightarrow P_1$ that is $\displaystyle \Delta w=\Delta w_a + \Delta w_b$

    via difference quotients it can write :

    $\displaystyle \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}$$\displaystyle = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h} + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h} $

    but $\displaystyle h_x=h*cos(\phi) $ and $\displaystyle h_y=h*sin(\phi) $

    then

    $\displaystyle \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}$ $\displaystyle = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{\frac{h_x}{cos(\phi)}} + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{\frac{h_y}{sin(\phi)}} $

    $\displaystyle \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}$ $\displaystyle = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi) + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi) $

    $\displaystyle \frac{\Delta w}{h} = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi) + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi) $

    $\displaystyle \frac{dw}{ds} $$\displaystyle = \lim_{h_x \rightarrow 0}{\frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi)} + \lim_{h_y \rightarrow 0}{\frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi)} $

    the first limits when $\displaystyle h_x \rightarrow 0 $ means $\displaystyle x_1 \rightarrow x_o $
    then the second difference quotient can be write :

    $\displaystyle \frac{f(x_o,y_o)}{ds} = f_x(x_o,y_o)*cos(\phi) + \lim_{h_y \rightarrow 0}{\frac{f(x_o, y_o+h_y) - f(x_o,y_o)}{h_y}*sin(\phi)} $

    and all thogether :

    $\displaystyle \frac{df(x_o,y_o)}{ds} = f_x(x_o,y_o)*cos(\phi) + f_y(x_o,y_o)*sin(\phi) $

    this is a wrong way ?
    Attached Thumbnails Attached Thumbnails Directional derivative definition via difference quotients-ddd.jpg  
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