Directional derivatives and the gradient vector problem

[zhuyilun]

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?

 

[LCKurtz]

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.

 

[zhuyilun]

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

 

[LCKurtz]

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.

 

[zhuyilun]

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

 

[LCKurtz]

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.

 

[zhuyilun]

i get it now, thank you so much