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Chapter 12: Problem 2

Describe the usual appearance of a smooth surface at a saddle point.

Short Answer

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Question: Describe the usual appearance of a smooth surface at a saddle point. Answer: At a saddle point, the surface appears to be bending in opposite directions, creating a shape that resembles a horse saddle or a Pringles chip. The surface at a saddle point is neither purely concave nor purely convex, but a combination of both, with positive curvature in one direction and negative curvature in the orthogonal direction. This results in a "valley" shape along one axis and a "hill" shape along the other.

Step by step solution

01

Define a saddle point

A saddle point is a point on a smooth surface where the surface is concave (curving upward) in one direction and convex (curving downward) in the orthogonal direction. In simpler terms, the surface curves upwards in one direction while curving downwards in the other, perpendicular direction.
02

Understand the appearance of a saddle point

At a saddle point, the surface appears to be bending in opposite directions, creating a shape that resembles a horse saddle or a Pringles chip. The surface at a saddle point is neither purely concave nor purely convex, but a combination of both.
03

Relate the saddle point to the shape of the surface

The particular appearance of a saddle point is a result of the varying curvature in different directions. As we move along the surface in one direction, the surface will appear to be curving upward, with a positive sign of curvature. Moving in the orthogonal direction, the surface will appear to be curving downward, with a negative sign of curvature.
04

Present an example

One of the most common examples of a saddle point is the surface defined by the function z = f(x, y) = x^2 - y^2. It has a saddle point at (0, 0) as the surface is concave along the x-axis (positive curvature) and convex along the y-axis (negative curvature). The surface has a "valley" shape along the x-axis and a "hill" shape along the y-axis.

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