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Math Help - Vector calculus.

  1. #1
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    Vector calculus.

    The temperature of points in space is given by T (r) = x2 + y2 z. A mosquitolocated at (1, 1, 2) desires to fly in such a direction that it cools as quickly aspossible. In which direction should it set out?
    Another mosquito is flying at a speed of 5 ms−1 in the direction of the vector(4, 4,
    −2). What is its rate of increase of temperature (with respect to time)as it passes through the point (1, 1, 2)?


    [IMG]file:///page1image6232[/IMG]
    1. (ii) A masochistic hiker wants to find the steepest route up a mountain of altitudeh(x,y) = 100−x2 −2y2. Assuming she starts at (x,y) = (√2,7) (i.e. h = 0)find the projection of the steepest path upon the xy plane.
      Hint: Show that the hiker should always move in a direction dr such thatdy/dx = 2y/x.
    2. (iii) Find the equation of the normal line and tangent plane to the surface z = 1+xyat the point (1, 1, 2).



    [IMG]file:///page1image8508[/IMG] [IMG]file:///page1image8592[/IMG]

    I have spent a lot of time working through this question. I assume that for the first one you use del to find the vector in the direction of maximum increase of the function and then take the negative to find the the direction where I would increase quickest.
    Thank you
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  2. #2
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    Re: Vector calculus.

    Yes, \nabla T points in the direction of fastest increase of T so that -\nabla T points in the direction fastest decrease. The mosquito should fly in the direction of -\nabla T. The rate of change of T in the direction of vector v is the dot product of \nabla T and a unit vector in the direction of v. What is a unit vector in the direcction of (4, 4, -2)?

    Yes, \nabla a points in the direction of fastest increase of a(x, y). Find \nabla (100- x^2- 2y^2) and evaluate it at [itex](\sqrt{2}, 7)[/tex].

    \nabla z points perpendicular to the surface. Evaluating at (1, 1, 2) gives the normal vector at that point.

    Now, you should know that a line in the direction of vector (A, B, C) though (x_0, y_0, z_0) is given by the parametric equations x= At+ x_0, y= Bt+ y_0, z= Ct+ z_0 and the tangent plane is given by A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0.
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  3. #3
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    Re: Vector calculus.

    For the last question would you make 1+xy-z=a constant to find the tangent and normal?
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