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    Gradients : Elliptic Paraboloid and Vector Fields

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    Suppose that a mountain has the shape of an elliptic paraboloid , where a and c are constants, x and y are the east-west and north-south map coordinates and z is the altitude above the sea level (x,y,z are measured all in metres). At the point (1,1), in what direction is the altitude increasing most rapidly? If a marble were released at (1,1), in what direction would it begin to roll?

    Let scalar differentiable functions and , (F, G vectors), differentiable vector fields. Show that

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    • Suppose that a mountain has the shape of an elliptic paraboloid
    ,
    where a and c are constants, x and y are the east-west and north-south map coordinates and z is the altitude above the sea level (x, y, z are measured all in metres). At the point (1, 1), in what direction is the altitude increasing most rapidly? If a marble were released at (1, 1), in what direction would it begin to roll?

    Solution:
    The purpose of this question is to allow us to understand the relationship between the directional derivative and the gradient ( ). To be more specific, we will be able to find the directions of the steepest ascent and descent from the ...

    Solution Summary

    Gradients are investigated with respect to an Elliptic Paraboloid and Vector Fields. The solution is detailed and well presented.

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