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

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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 ...

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