Directional derivatives and the gradient vector problem

Click For Summary
SUMMARY

The discussion centers on the problem of demonstrating that pyramids formed by tangent planes to the surface defined by the equation xyz=1 in the first octant have equal volumes. Participants are guided to derive the equation of the tangent plane at a point (a, b, c) on the surface and to calculate the volume of the resulting tetrahedron formed by this plane and the three coordinate planes. The key takeaway is that the volume remains constant for any point (a, b, c) on the surface within the first octant.

PREREQUISITES
  • Understanding of multivariable calculus concepts, specifically tangent planes.
  • Familiarity with the equation of the surface xyz=1.
  • Knowledge of volume calculation for tetrahedrons.
  • Basic comprehension of coordinate geometry in three dimensions.
NEXT STEPS
  • Study the derivation of tangent planes in multivariable calculus.
  • Learn how to calculate the volume of a tetrahedron given its vertices.
  • Explore the implications of the implicit function theorem in relation to surfaces.
  • Investigate the geometric interpretation of gradients and directional derivatives.
USEFUL FOR

Students of multivariable calculus, educators teaching advanced calculus concepts, and anyone interested in geometric interpretations of mathematical surfaces and their properties.

zhuyilun
Messages
26
Reaction score
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?
 
Physics news on Phys.org
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.
 
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
 
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.
 
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
 
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
 
i get it now, thank you so much
 

Similar threads

Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K
Directional Derivative at an Angle with a 3d Gradient
  • · Replies 4 ·
Replies
4
Views
3K
Problem with a directional derivative calculation
  • · Replies 6 ·
Replies
6
Views
2K
Meaning of zero gradient vector with existant directional vector
  • · Replies 12 ·
Replies
12
Views
9K
What is the Maximum Rate of Change in Scalar Fields?
  • · Replies 4 ·
Replies
4
Views
2K
Directional Derivative at an Angle from the Gradient
  • · Replies 8 ·
Replies
8
Views
5K
Vector analysis problem about a gradient
  • · Replies 13 ·
Replies
13
Views
1K
Undergrad Gradient vectors and level surfaces
  • · Replies 3 ·
Replies
3
Views
2K
Significance of nonvanishing gradient
  • · Replies 22 ·
Replies
22
Views
4K
Can you find the gradient of a vector?
  • · Replies 6 ·
Replies
6
Views
3K