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Math Help - PROOF of the Directional Derivative

  1. #1
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    PROOF of the Directional Derivative

    What's the PROOF of the Directional Derivative of a 3-variable function?
    DuF(x,y,z)=(Fx,Fy,Fz).(U1,U2,U3) where U=(U1,U2,U3) is a unit vector.
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  2. #2
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    Assuming that F is differentiable at the given point, (x_0, y_0. z_0) then it can be approximated by the linear function F_x(x_0,y_0,z_)(x- x_0)+ F_y(x_0,y_0,z_0)(x-y_0)+ F_z(x_0,y_0,z_0)(x- x_0)+ F(x_0,y_0,z_0). (That follows from the definition of "differentiable at a point")

    The line through (x_0, y_0, z_0), in the direction of U= (U_1, U_2, U_3), is given by the parametric equations x= x_0+ U_1t , y= y_0+ U_2t, z= z_0+ U_2t.

    To find the derivative of F in the direction U, replace x, y, and z with those in the approximation for F, to get F_x(U_1t)+ F_y(U_2t)+ F_z(U_3t)=  (F_xU_1+ F_yU_2+ F_zU_3)t, a linear function of t with derivative equal to its slope, F_xU_1+ F_yU_2+ F_zU_3.
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