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Math Help - compute the directional derivatives

  1. #1
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    compute the directional derivatives

    Hi!
    There is an exam on Tuesday.
    I have no idea of these problems.
    Please help~~ Thank you^^

    Compute the directional derivatives of the following functions in the directions indicated.

    (a) x+xy-xz+2y at (2,1,0) in the direction in which the directional derivative is a maximum.

    (b) e^xsiny (as a function of two variables) at (0,∏/6) in the direction making a +60˚ angle with the positive x-axis.

    (c) x+2y-3x+y at (-2,4) in the direction of the tangent (from left to right)of the curve discribed by y=x

    (d) e^xyz+x+y at (1,1,1) in the direction of the curve described by x=t, y=2t-1, z=t

    (e) x-y+z in the direction of the outer normal of ghe surface x+y+z=7 at (2,-1,2)
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  2. #2
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    If we have a scalar field (acts like a function) \phi(x_n,y_n,z_n)=\phi(P_n)\,,\,P_n=(x_n,y_n,z_n), we define its gradient as

    \nabla \phi(P_n)=\frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y} \hat{j} + \frac{\partial \phi}{\partial z}\hat{k}.

    The gradient will always point in the direction where the function increases the most.

    The directional derivative in the direction of \vec{u} then becomes

    D_u \phi(P_n)=\nabla \phi (P_n)\cdot \vec{u}

    where \vec{u} is a unit vector.


    Now you just have to compute \frac{\vec{v}}{|\vec{v}|}=\vec{u} for all the problems below.
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  3. #3
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    For (b), a unit vector in the direction making angle \theta with the positive x-axis is cos(\theta)\vec{i}+ sin(\theta)\vec{j}.
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  4. #4
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    For this solution, what is the vector u in problems (d) and (e) ??
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  5. #5
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    (d) e^xyz+x+y at (1,1,1) in the direction of the curve described by x=t, y=2t-1, z=t
    What is the tangent vector to this curve at (1, 1, 1)?

    (e) x-y+z in the direction of the outer normal of the surface x+y+z=7 at (2,-1,2)
    What is the normal vector to this surface at (2, -1, 2)?
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