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1 A Two-Dimensional Comparison of Differential Calculus and Differential Geometry Andrew Grossfield, Ph.D Vaughn College of Aeronautics and Technology Abstract and Introduction: Plane geometry is mainly the study of the properties of polygons and circles. Differential geometry is the study of curves that can be locally approximated by straight line segments. Differential calculus is the study of functions. These functions of calculus can be viewed as single-valued branches of curves in a coordinate system where the horizontal variable controls the vertical variable. In both studies the derivative multiplies incremental changes in the horizontal variable to yield incremental changes in the vertical variable and both studies possess the same rules of differentiation and integration. It seems that the two studies should be identical, that is, isomorphic. And, yet, students should be aware of important differences. In differential geometry, the horizontal and vertical units have the same dimensional units. In differential calculus the horizontal and vertical units are usually different, e.g., height vs. time. There are differences in the two studies with respect to the distance between points. In differential geometry, the Pythagorean slant distance formula prevails, while in the 2- dimensional plane of differential calculus there is no concept of slant distance. The derivative has a different meaning in each of the two subjects. In differential geometry, the slope of the tangent line determines the direction of the tangent line; that is, the angle with the horizontal axis. In differential calculus, there is no concept of direction; instead, the derivative describes a rate of change. In differential geometry the line described by the equation y = x subtends an angle, α, of 45° with the horizontal, but in calculus the linear relation, h = t, bears no concept of direction. Often the three words, derivative, slope and rate of change are used interchangeably which a perceptive student might find confusing. There are three important local properties of points on a trajectory in differential geometry: (1) position, (2) direction and (3) turning or curvature. In differential geometry, local curvature is dα described by the change in angle with differential distance along the curve, . It will be shown ds that the reciprocal of the curvature is the radius of curvature. On the other hand, differential calculus has no concepts of curvature and radius of curvature. Instead, in differential calculus, turning is described by the rate of change of the rate of change, which is the rate of change of velocity or the second derivative of the vertical variable. This paper aims to organize these concepts and thereby reveal their similarities and differences. Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright 2020, American Society for Engineering Education 2 Differential Geometry In differential geometry, the vertical and horizontal variables, x and y, have the same dimensional units: 1) The graph of the equation y = + √1 − x2 will be the upper half of a circle of radius 1. 2) The graph of the equation y = x will be a straight line in the direction of 45 ° to the horizontal and a slope of 1. 3) The direction of a straight line can be measured by using either α, the angle with the horizontal or m, the slope of the line. Note: y – y0 = m(x – x0) or Δy = m Δx. 4) Small changes in distance along the curve can be computed by the Pythagorean slant distance formula: ds = + √dx2 + dy2 . 5) The curves under consideration can be viewed as geometric shapes, and 6) the angle between intersecting curves will be measured by the difference in angle between the lines of tangency at the point of intersection. Additionally, the words slope or direction appropriately can be applied to 1st derivative. Rate of change might also be used. In this case, every calculation must conform to geometric facts. The conventional differential geometry course extends the concepts of the 2-dimensional case to spaces of dimension greater than two. Differential Calculus In differential calculus, the variables in the describing equations represent quantities whose units are usually different. For example, say the vertical variable, y, is the signed distance from the horizontal axis and the horizontal variable is time, t. If a ball is thrown in the air, the relationship between y and t can be viewed as a curve in the y, t space. However, the concept of distance between two points on this curve has no meaning. The curve has a tangent line at each point, but the concept of direction of this tangent line also has no meaning. Figure 1 Rate of change Figure 2 Turning up and turning down Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright 2020, American Society for Engineering Education 3 Figure 3 Changes in amplitude and time lags of sinusoids Instead, we can say the tangent line is rising or falling and focus on the rate of rise or fall using the rate, r, as a measure of the rate of rise in analogy with the 2-dimensional differential geometric case as shown in figure 1: y – y0 = r(t – t0) or Δy = r Δt. The terms rate of change and velocity are appropriately applied to the 1st derivative but the term slope may be inappropriate. When two sinusoidal voltages of time are graphed, there can be a time lead or lag between the peaks and a size difference between the amplitudes, but a concept of slant distance between the peaks is meaningless as shown in figure 3 above Point, Interval and Global Properties of Functions and Curves The point or “local” properties are features of a curve that pertain to a feature at a single point. The three important local properties of ordinary points are the position, the direction and the rate of turn. Interval properties of a curve are the features of a curve that prevail between the endpoints of the interval such as a curve may be positive or may be rising in the interval with endpoints a and b. A function that is positive on the interval (a, b) would be written as f(x) > 0 for a < x < b. The area between the x-axis and a positive function f(x) above the horizontal interval (a, b) could b be written as A{f(x); a, b } = f(x)dx. ∫a Global properties of the curve pertain to the entire curve and are not limited to an interval. The following table compares these properties. Point property A curve rises at the point a0 if the slope of the tangent line at a0 is positive. Global property A curve such as y = e x is monotonic if it rises at every point, x, < –∞ < x < ∞. Interval property A wiggling curve that rises from a low at x = a to the next maximum at x =b is said to rise monotonically on the interval between a and b and would be written as f′(x) > 0 for a < x < b. Similarly, a curve that falls from a high at x = c to the next minimum at x = d is said to fall monotonically on the interval c < x < d. Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright 2020, American Society for Engineering Education 4 Figure 4. Local behavior of the circle in each of the 4 quadrants The circle x2 + y2 = 16, shown in figure 4 can provide graphical or visual examples of the three important point properties of a continuous, well-behaved curve. In the 1st quadrant: y > 0 , y′ < 0 and y′′ < 0. In the 2nd quadrant: y > 0 , y′ > 0 and y′′ < 0. In the 3rd quadrant: y < 0 , y′ < 0 and y′′ > 0. In the 4th quadrant: y < 0 , y′ > 0 and y′′ > 0. The previous discussion leads to the conclusion that both differential calculus and differential geometry are studies of the local properties of well-behaved curves. Both studies can be encapsulated in the three concepts: position, direction, or rate of change and turning. Imagine you are moving along a trajectory in a 2-dimensional space: 1) Where are you? 2) Where are you heading? and 3) When the direction varies, how are you deviating from the tangent line? At a fixed point on a trajectory, both the curve and the tangent line have the same position and the same tilt. The tilt is obtained from the 1st derivative which provides either the direction or rate of rise: items 1 and 2 above. But item 3, the curvature or turning is related to variations in the tilt or direction of the curve. Curvature is deviation from the tangent line as shown in figures 1, 2 and 4. We begin by recognizing that there are two ways to measure the tilt of a line: α, the angle with the horizontal and the rate of change or 1st derivative of the line. We also recognize that if either the angle, α or the slope, m is known the other can be computed by applying either the relationship m = tan(α) or the inverse relationship α = arctan(m). The question before us is: “Should the curvature or turning be measured using the angle or the rate of change?” The answer is that the angle is used in 2-dimensional differential geometry but the rate of change is used in differential calculus. Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright 2020, American Society for Engineering Education 5 We should recognize that when the trajectory is straight, the tilt is constant and the 2nd derivative of the vertical variable is zero.
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