Lec 32 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins the lecture by discussing the importance of MIT OpenCourseWare and its reliance on donations. The main topic of the lecture is parametric curves, specifically the concept of arc length. The professor reviews the parametric representation of a circle and introduces the formula for arc length in parametric form, which involves taking the square root of the sum of the squares of the derivatives of x and y with respect to the parameter t. The lecture also covers how to find the rate of change of arc length with respect to t, which can be interpreted as the speed of a particle moving along the curve. The constant speed of a particle moving counterclockwise along the circle is highlighted, and the idea of changing this speed by introducing a scaling factor is also discussed.

This paragraph delves deeper into the mathematical rigor behind the arc length formula. The professor explains the transition from the approximate calculation of (delta s)^2 as (delta x)^2 + (delta y)^2 to the limit definition involving derivatives. The importance of dividing by a nonzero quantity before taking the limit is emphasized to ensure mathematical validity. The professor also warns against the misuse of notation, particularly with squared differentials and ratios, and clarifies the correct usage with examples. The difference between the arc length formula and the second derivative is highlighted, with the latter involving a dt^2 term in the denominator. The discussion serves to clarify the precise mathematical manipulations involved in working with parametric curves.

The professor presents a more complex example involving a non-constant speed parameterization of an ellipse. The parametric equations x = 2 sin t and y = cos t are given, and the professor guides through the process of deriving the rectangular equation for this curve, which is found to be an ellipse. The starting point and direction of the curve are discussed, with the curve traced clockwise starting from the point (0, 1). The speed of the particle is calculated based on the derivatives of x and y with respect to t. However, the arc length integral for a complete traversal of the ellipse is identified as a non-elementary integral, meaning it cannot be expressed in terms of standard functions and requires numerical methods or approximations.

The lecture shifts focus to the calculation of surface area, specifically the surface area of an ellipsoid formed by revolving the previously discussed ellipse around the y-axis. The professor explains the need for a parameterization that includes both the arc length element ds and the distance from the curve to the axis of rotation, identified as x in this case. The integral for surface area is set up, involving the product of 2 pi, the distance x (which is 2 sin t), and the differential arc length ds. The limits of integration are discussed, with the professor highlighting the importance of understanding the range of the parameter t that corresponds to a complete revolution around the y-axis, which is from 0 to pi for the ellipsoid's surface area.

The professor addresses several questions from students regarding the parametric representations and the integration process. The first question concerns the plotting of the ellipse without considering the parameter t, to which the professor explains that the parameter t is essential for depicting the entire curve and that it should be thought of as a trajectory over time. Another question asks why the integration for arc length is done with respect to t rather than x, prompting the professor to clarify that when dealing with curves, one can choose any variable for integration, and in this case, t is the natural choice. The professor emphasizes the need to let go of the traditional y = f(x) perspective and instead consider x and y as functions of another variable, t.

The lecture concludes with an introduction to polar coordinates, a different system for describing points in a plane based on distance from the origin and an angle from the horizontal axis. The professor presents the fundamental formulas for converting between Cartesian (x, y) and polar (r, theta) coordinates: x = r cos(theta) and y = r sin(theta). These equations are the starting point for all other relationships in polar coordinates. The professor also introduces the concept of the radius r as the square root of x^2 + y^2 and the angle theta as the inverse tangent of y/x, noting that these latter expressions are subject to ambiguities and must be used with caution.

The professor continues the discussion on polar coordinates, emphasizing the unambiguous definitions of r and theta in terms of x and y. The potential ambiguities in the formulas for r and theta are highlighted, particularly the issues with negative values and the inverse tangent function. The lecture then moves on to examples of polar coordinates, illustrating how points in the Cartesian plane can be represented in polar coordinates with multiple valid representations due to the nature of the angle theta. The professor also discusses the typical conventions for the ranges of r and theta in polar coordinates, noting that while there are common ranges, they are not always consistent and must be carefully considered.

The professor provides more advanced examples to further illustrate the use of polar coordinates. This includes the translation of simple rectangular equations like y = 1 into polar form, resulting in r = 1 / sin(theta). The importance of understanding the range of theta for each polar equation is emphasized, as it is crucial for integration purposes. The lecture also touches on the representation of off-center circles in polar coordinates, using the equation (x-a)^2 + y^2 = a^2 as an example. The professor demonstrates how to convert this equation into polar form and discusses the significance of the range of theta for this representation, which is found to be -pi/2 < theta < pi/2.