# Directional derivatives for altitude

• reddawg
In summary, to maintain a constant altitude at the point (-100,-100,360) on a hill with the shape of the graph z=500-0.006x2-0.008y2, you need to find a gradient vector ∇f and a vector u such that the directional derivative is zero. This can be achieved by setting u to <1, -0.75> or any other solution that is a unit vector.
reddawg

## Homework Statement

Suppose you are standing at the point (-100,-100,360) on a hill that has the shape of the graph of z=500-0.006x2-0.008y2.

In what direction should you head to maintain a constant altitude?

## Homework Equations

Duf = ∇f$\bullet$u
formula for directional derivative

## The Attempt at a Solution

Here's my idea. If the directional derivative at the point specified is zero, then that's equivalent to maintaining constant altitude. Therefore, I must find the corresponding gradient vector ∇f when the directional derivative is zero right? I have no idea how to carry this out however.

reddawg said:

## Homework Statement

Suppose you are standing at the point (-100,-100,360) on a hill that has the shape of the graph of z=500-0.006x2-0.008y2.

In what direction should you head to maintain a constant altitude?

## Homework Equations

Duf = ∇f$\bullet$u
formula for directional derivative

## The Attempt at a Solution

Here's my idea. If the directional derivative at the point specified is zero, then that's equivalent to maintaining constant altitude. Therefore, I must find the corresponding gradient vector ∇f when the directional derivative is zero right? I have no idea how to carry this out however.

Yes, find the gradient vector. But you can't choose what it is, it's fixed by the problem statement. What you need to find is a vector u such that the directional derivative is zero.

0 = ∇f$\bullet$u

You say ∇f is fixed:

∇f = <zx,zy>

= < -0.012x, -0.016y >

= < 1.2, 1.6 > Subs. coordinates given

Then solve for u?

reddawg said:

0 = ∇f$\bullet$u

You say ∇f is fixed:

∇f = <zx,zy>

= < -0.012x, -0.016y >

= < 1.2, 1.6 > Subs. coordinates given

Then solve for u?

Yes, write u=<ux, uy>. Since the dot product of u with the gradient is 0 you should be able to find a relation between ux and uy. You won't be able to find a unique u. But the question is only asking for a direction.

So u could just equal < 0 , 0 > ?

reddawg said:
So u could just equal < 0 , 0 > ?

It could. That means you just stay in the same place and don't go anywhere. But the question is asking for a direction in which you could move and not change altitude. Find another solution.

Oh I get it.

I'll use u = < 1 , -0.75 > , that comes out to zero.

Thanks Dick.

reddawg said:
Oh I get it.

I'll use u = < 1 , -0.75 > , that comes out to zero.

Thanks Dick.

Right and you're welcome. Actually there are lots of solutions but they all point in either that direction or the opposite direction.

u is supposed to be a unit vector. So take your u and divide by its magnitude.

Chestermiller said:
u is supposed to be a unit vector. So take your u and divide by its magnitude.

Or they might want an angular bearing. Hard to say. reddawg might know based on previous questions.

## 1. What is a directional derivative for altitude?

A directional derivative for altitude is a mathematical concept used in the field of geosciences to measure the rate of change of altitude along a specific direction. It is often used in terrain modeling and topographic mapping.

## 2. How is a directional derivative for altitude calculated?

The directional derivative for altitude is calculated by taking the dot product of the gradient of altitude and a unit vector in the desired direction. This results in a scalar value that represents the rate of change of altitude in that direction.

## 3. Why is a directional derivative for altitude important?

A directional derivative for altitude is important because it allows scientists to analyze the steepness and changes in altitude along a specific direction. This information is useful in understanding the topography and landscape of an area, as well as predicting potential hazards such as landslides or avalanches.

## 4. What are some real-world applications of directional derivatives for altitude?

Directional derivatives for altitude are used in various fields such as geology, geography, environmental science, and civil engineering. They are used in terrain analysis, slope stability analysis, and in creating accurate topographic maps for navigation and planning purposes.

## 5. Are there any limitations to using directional derivatives for altitude?

While directional derivatives for altitude provide valuable information about the terrain, they do have limitations. They assume a smooth and continuous terrain, and do not take into account smaller features such as rocks or trees. Additionally, they may be affected by errors in elevation data or the chosen direction of analysis.

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