# Gradients : Elliptic Paraboloid and Vector Fields

Please see the attached file for the fully formatted problems.

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

Â© BrainMass Inc. brainmass.com November 24, 2022, 11:58 am ad1c9bdddfhttps://brainmass.com/math/vector-calculus/gradients-elliptic-paraboloid-vector-fields-33378

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

â€¢ 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.