Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Prove that the principal unit normal vector \(\mathbf{N}\) points toward the concave side of a plane curve.

Short answer

The statement is False. The principal unit normal vector \(\mathbf{N}\) does not necessarily point toward the concave side of a plane curve, predominantly because the notions of 'concave' and 'convex' are not applicable for plane curves.

Step-by-step solution

Understanding the statement

It is asked to prove that the principal unit normal vector \(\mathbf{N}\) points toward the concave side of a plane curve. However, it's important to note that the concept of 'concavity' and 'convexity' are not usually applicable to plane curves. These are used in the context of curves in the Cartesian plane.

Verifying the statement

The direction in which the principal unit normal vector \(\mathbf{N}\) points is independent of whether it’s towards the concave side of a plane curve. Because, in the context of plane curves, there are no 'concave' or 'convex' sides, instead the curve could turn left or right.

Finding counterexample

Consider a circle with its center at the origin in the xy-plane. The normal vector at each point on the circle's circumference points towards the center of the circle. If the circle is considered to be a 'plane curve', it is not possible to identify a concave or convex side towards which the normal vector would point. This shows that the provided statement is untrue.

Key concepts

Plane Curve Concavity

In calculus, understanding the concavity of plane curves is vital for visualizing and analyzing the path that a curve takes. When we speak of a plane curve's concavity, we're referring to the direction that the curve bends. A curve is said to be concave upward (or concave to the sky) when it bends such that it could contain water, like a cup. Conversely, it's concave downward (or concave to the ground) if it bends in a way that water would spill out, resembling an arch.

Mathematically, the concavity is determined by the sign of the second derivative of the curve's equation. Positively, this indicates concave upward, and negatively, concave downward. For instance, the function with equation y=x^2 is concave upward, as its second derivative y''=2 is always positive.

Normal Vector in Calculus

A normal vector in calculus represents a line or vector that is perpendicular to the tangent vector at a particular point on a curve. In the context of a two-dimensional plane curve, the principal unit normal vector, often denoted as \( \mathbf{N} \), is key because it provides significant information about the curve's geometry and direction of curvature.

To construct this normal vector, one must first find the tangent vector to the curve at the point of interest. The principal unit normal vector is then a vector that is perpendicular to this tangent vector and has a magnitude of one. It is usually obtained by taking the derivative of the unit tangent vector with respect to the arc length parameter of the curve, and then normalizing it.

Proofs in Calculus

Proofs in calculus are logical arguments that establish the truth of mathematical statements. These proofs can range from simple, direct computations to complex, abstract concepts requiring a great deal of insight. To successfully prove a statement in calculus, several steps are commonly involved: clearly stating the theorem or property to be proved, making precise any definitions, constructing a logical sequence of assertions based on known theorems and axioms, and connecting these assertions to conclude the proof.

For example, a proof that the principal unit normal vector \( \mathbf{N} \) points toward the concave side of a curve would hinge upon defining concavity, curvature, and the normal vector itself. From there, one would need to demonstrate that these definitions and known properties support the statement's truth. As noted in the solution steps, one must be careful with the characteristics of plane curves when attempting to prove statements specifically about them.

Concave and Convex Curves

The distinction between concave and convex curves is fundamental in the study of the geometry of curves. A curve is concave if, at any point on the curve, any line segment joining two points on the curve will lie below or on the curve itself. In contrast, a curve is convex if any such line segment remains above or on the curve. This characteristic reflects the overall 'bowl'-like or 'dome'-like shape of the curve, respectively.

When assessing the concavity or convexity of a curve, one typically looks at the behavior of the first and second derivatives. For a function y=f(x), if the first derivative f'(x) is increasing (second derivative f''(x) is positive), the curve is concave up; if f'(x) is decreasing (second derivative f''(x) is negative), the curve is concave down.

The relevance of these concepts to the normal vector is that the normal vector can be used to determine the direction of concavity. However, this relationship is not as straightforward when dealing with plane curves that don't have an inherent 'concave' or 'convex' side, illustrating the importance of understanding the specific context within which these terms are used.