 Hello and welcome to this screencast on section 9.8, arc length and curvature. This screencast is going to cover curvature. For a smooth space curve, the curvature measures how fast the curve is bending or changing direction at a given point. For example, we expect that a line should have zero curvature everywhere, while a circle, which is bending the same at every point, should have constant curvature. Circles with larger radii should have smaller curvatures. This is because the larger the radius, the less sharp the bend is on a circle. To measure the curvature, we first need to describe the direction of the curve at a point. We may do this using a continuously varying tangent vector to the curve, as shown on the left in the figure here. The direction of the curve is then determined by the angle phi each tangent vector makes with a horizontal vector, as shown on the right side. Informally speaking, the curvature will be the rate at which the angle phi is changing as we move around the circle. This rate of change depends on how we move along the curve. If we move with a greater speed, then phi will change more rapidly. This is why the speed limit is sometimes lowered when we enter a curve on the highway. To eliminate the dependence on speed, we choose to work with an arc length parameterization of the curve, which means we move along the curve with unit speed. We are now ready to state the formal definition of curvature for a vector valued function r. As noted on the previous slide, to eliminate the dependence on speed, we use the arc length parameterization. Then the tangent vectors, capital T, are given by the derivative of r and the definition of curvature. If r is a smooth space curve and s is the arc length parameter for r, then the curvature, kappa of r, is given by the magnitude of the derivative of capital T. Given a function in terms of an arbitrary parameter T, it can sometimes be difficult to find the arc length parameterization as we used on the previous slide in the definition. Thus, it is helpful to have alternative formulas for curvature that do not depend on arc length, and we give two such formulas here. If r is a smooth space curve, r prime is not zero and the second derivative of r exists, then the curvature, denoted by kappa, satisfies either of the given formulas here. We may utilize either formula and we may find ourselves in situations where one formula is easier to apply than the other. Note here that the first formula relies on capital T, the unit tangent vector.