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Curvature is a way to measure how much a curve deviates from being a straight line—or, in simpler terms, how "curvy" it is. For a plane curve defined by a vector function \( \mathbf{r}(t) \), the curvature, denoted \( \kappa \), quantifies how sharply the curve bends at a given point. Calculating curvature involves several steps:

• First, find the tangent vector \( \mathbf{T}(t) \), which is the unit vector in the direction of the derivative \( \mathbf{r}'(t) \).
• The curvature is given by \( \kappa = \frac{| \mathbf{T}'(t) |}{| \mathbf{r}'(t) |} \).

This formula considers the rate of change of the tangent vector's direction with respect to the arc length of the curve. This is critical in ensuring the curvature measures only how the direction changes, not how fast we move through the curve.

The unit normal vector, \( \mathbf{N}(t) \), is a vector that is perpendicular to the tangent vector of a curve at a given point. It provides significant insights into the direction in which the curve is bending at that specific point.To find the unit normal vector, follow these steps:

• First, compute the derivative of the tangent vector, \( \mathbf{T}'(t) \).
• Normalize \( \mathbf{T}'(t) \) by dividing it by its magnitude to ensure it has a length of 1: \( \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} \).

By doing this, you obtain the unit normal vector which points towards the center of the curve's osculating (curvature-defining) circle, helping us understand the inherent geometry of the curve better.

The derivative is a fundamental concept in calculus that measures the rate of change of a function. For a given vector function \( \mathbf{r}(t) \), the derivative, \( \mathbf{r}'(t) \), provides us with the velocity vector, indicating the direction and rate at which the position vector is changing.When dealing with curves, the derivative helps in finding:

• The velocity of a moving point on the curve.
• The tangent vector by normalizing the velocity vector.

In our exercise, finding the derivative of \( \mathbf{r}(t) \) involves differentiating each component separately. Given \( \mathbf{r}(t) = (\cos t + t \sin t) \mathbf{i} + (\sin t - t \cos t) \mathbf{j} \), the derivative becomes \( \mathbf{r}'(t) = t \cos t \mathbf{i} + t \sin t \mathbf{j} \). This results in a simpler expression, reflecting the instantaneous rate at which the position on the curve changes, and forms the first step in understanding further properties like the tangent vector, curvature, and unit normal vector.