Directional derivatives and the gradient vector problem

In summary: Here's a picture showing just one example of a plane tangent to your surface. The coordinate planes outlined in red give the other faces.In summary, to solve this problem, you need to find the equation of the tangent plane at a point on the surface xyz=1 in the first octant, and then calculate the volume of the pyramid formed by this tangent plane and the three coordinate planes. This volume should be the same for any point on the surface in the first octant.
  • #1
zhuyilun
27
0

Homework Statement


show that the pyramids cut off from the first octant by any tangent planes to the surface xyz=1 at points in the first octant must all have the same volume


Homework Equations





The Attempt at a Solution



i don't know how to start this problem. any hints?
 
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  • #2
Start by writing the equation of the tangent plane at a point (a,b,c) on the surface in the first octant. Then finish by calculating the mentioned volume.
 
  • #3
LCKurtz said:
Start by writing the equation of the tangent plane at a point (a,b,c) on the surface in the first octant. Then finish by calculating the mentioned volume.

i don't know what it means by "the surface in the first octant", what should the general equation look like? thank you
 
  • #4
zhuyilun said:
i don't know what it means by "the surface in the first octant", what should the general equation look like? thank you

xyz = 1 is the equation of a surface. If (a, b, c) is a point on the surface in the first octant, you can calculate the equation of the tangent plane to the surface at that point. That tangent plane and the three coordinate planes make the sides of a pyramid (tetrahedron). Calculate its volume. The problem is to show that the answer you get doesn't is the same for any (a,b,c) on the surface in the first octant.
 
  • #5
LCKurtz said:
xyz = 1 is the equation of a surface. If (a, b, c) is a point on the surface in the first octant, you can calculate the equation of the tangent plane to the surface at that point. That tangent plane and the three coordinate planes make the sides of a pyramid (tetrahedron). Calculate its volume. The problem is to show that the answer you get doesn't is the same for any (a,b,c) on the surface in the first octant.

i am sorry, but what do you mean by " three coordinate planes". and can you explain a little bit more about how to find sides of the pyramid
 
  • #6
zhuyilun said:
i am sorry, but what do you mean by " three coordinate planes". and can you explain a little bit more about how to find sides of the pyramid

Here's a picture showing just one example of a plane tangent to your surface. The coordinate planes outlined in red give the other faces.

pyramid.jpg
 
  • #7
i get it now, thank you so much
 

Related to Directional derivatives and the gradient vector problem

1. What is a directional derivative?

A directional derivative is a measure of how much a function changes in a specific direction at a given point. It is represented by the slope of a tangent line to the function at that point in the specified direction.

2. How is a directional derivative calculated?

A directional derivative is calculated by taking the dot product of the gradient vector of the function and a unit vector in the specified direction.

3. What is the purpose of finding directional derivatives?

Finding directional derivatives allows us to understand how a function changes in a specific direction, which can be useful in applications such as optimization and modeling physical systems.

4. What is the gradient vector problem?

The gradient vector problem is a mathematical problem that involves finding the gradient vector of a function at a given point. It is often used in multivariable calculus to find the direction of steepest ascent or descent of a function.

5. How is the gradient vector problem solved?

The gradient vector problem is typically solved by taking the partial derivatives of the function with respect to each variable and then combining them into a vector. This vector represents the direction of maximum change of the function at the given point.

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