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A Two-Dimensional Comparison of Differential and Differential

Andrew Grossfield, Ph.D Vaughn College of Aeronautics and Technology

Abstract and Introduction:

Plane geometry is mainly the study of the properties of and . Differential geometry is the study of that can be locally approximated by straight segments. is the study of functions. These functions of calculus can be viewed as single-valued branches of curves in a where the horizontal variable controls the vertical variable.

In both studies the 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 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 line determines the direction of the tangent line; that is, the 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) , (2) direction and (3) turning or . In differential geometry, local curvature is dα described by the change in angle with differential distance along the , . It will be shown ds that the reciprocal of the curvature is the .

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 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 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 , and 6) the angle between intersecting curves will be measured by the difference in angle between the lines of tangency at the 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 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 . 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. If the curve turns upward, above the tangent line the tilt of the trajectory increases and if the curve turns down below the tangent line, the tilt decreases.

If height vs. time is being observed in differential calculus, the rate of change of height is called dy velocity and is measured as the 1st derivative, . When the velocity is not constant, the rate of dt changing velocity is called acceleration and is measured as the rate of change of the rate of d dy d2y change, the 2nd derivative, ( ) = . dt dt dt2

However, in the case of 2-dimensional differential geometry the rate of turn is not measured as dα the derivative of angle with respect to the horizontal variable, x. Geometric features are dx intrinsic, that is, independent of an external coordinate system. Rate of turn is measured as the dα rate of change of angle with respect to distance along the curve, . ds

dα On a straight line, the direction is constant and the change in angle with respect to arc , ds dα is zero. In this case of 2-dimensional differential geometry, turning is measured with which ds is symbolized by the Greek letter κ. The curvature, κ, indicates the rate of deviation from the dα straight tangent line. dα = ds. The curvature, κ is an intrinsic property of a point on the curve, ds meaning it is the same in all coordinate systems.

ds dα –1 dα 1 On a circle, ds = R dα and therefore, R = = ( ) and therefore = κ = . The dα ds ds R curvature is the reciprocal of the radius of the osculating circle as shown below in figure 5.

Figure 5 The change in equals the radius times the change in angle

In figure 6 below two ellipses are plotted together with their circles of curvature at their vertices. It is seen that the are greatest and least at the vertices.

Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright  2020, American Society for Engineering Education 6

Figure 6 The circles of curvature at the vertices of two ellipses

The following table summarizes and compares the local properties of the two studies.

2-Dimensional General Differential Calculus Differential Geometry with Different Horizontal and Vertical Dimensional Units

Position y(x) The original function position y(t) dy dy Direction α = arctan( ) The first derivative rate of change dx dt dα d2y Curvature = κ The second derivative acceleration ds dt2

dα As examples, contrast two curves. On a circle, the curvature, κ = remains constant as the arc ds d2y length, s, varies but varies. On the second curve, a vertical geometric parabola, y(x) = x2, dx2 2 d y dα is constant, but the curvature, , varies, being greatest at the and decaying to zero as dx2 ds x increases beyond bound.

Note 1: The equation κ = R-1 identifies the curvature of a circle with the reciprocal of its radius. A smaller circle has a greater curvature or rate of turn than a larger circle.

Note 2: As an application in , since force is proportional to acceleration, a pilot of a supersonic craft undergoes a higher physical stress when executing a tighter turn.

How Does the Process of Integration Compare in the Two Studies?

In differential geometry, the of a positive function over an interval, [a, b], will provide b the between the function and the x-axis over the interval and is written A = f(x) dx . ∫a

Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright  2020, American Society for Engineering Education 7

On the other hand in differential calculus, there is no concept of area. But the rate of change of a variable can be accumulated or integrated to provide the total change; e.g., say, we are interested in accumulating the varying charge, Q(t), flowing into a capacitor. The rate of change of charge is the through the capacitor, i(t). The total accumulated charge between times t1 and t2 is t obtained by the classic integration process Q =∫ 2 i(t) dt . No harm is done interpreting the t1 accumulated charge as the area under the graph of i(t), if it assists a student’s comprehension, but accumulated charge is not an area in the strict sense of differential geometry.

Extensions to Three Dimensional Curves

Similar concerns arise in 3-dimensional space curves that were confronted in the study of planar, 2-dimensional curves. Again the symbol κ is used to symbolize geometric deviation from the straight line of tangency but space curves like the spiral can twist into 3-dimensional space and the Greek letter, τ, is used to symbolize torsion or geometric deviation from being a planar curve. In 3-dimensional differential geometry, a curve would be described in parametric form as a x(s) vector function of arc length, s, along a curve, 퐑⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) = [y(s)] . As in the 2-dimensional case the z(s) x′(s) 퐝퐑⃗⃗⃗⃗⃗ unit tangent vector, 퐓⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) , is obtained as the vector derivative: 퐓⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) = (퐬) = [y′(s)] . Again 퐝퐬 z′(s) as in the 2-dimensional case, the derivative of 퐓⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) is normal to 퐓⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) and has a magnitude equal x′′(s) 퐝퐓⃗⃗⃗⃗⃗ to the curvature, κ, at the point described by the parameter, s: (퐬) = [y′′(s)] = κ 퐍⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) . The 퐝퐬 z′′(s) cross product of the unit vectors, 퐓⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) and 퐍⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) , yields a third unit vector, 퐁⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) , which completes a 3-dimensional orthonormal coordinate system, or frame, named after Frenet and Serret that moves along the curve as the parameter, s varies. When a curve is two dimensional, 퐝퐁⃗⃗⃗⃗⃗ the unit vector, 퐁⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) , remains to itself and = 0. The magnitude of the derivative of 퐁⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) 퐝퐬 is the torsion, τ. The twist out of the T-N plane is determined as the magnitude of the vector derivative of 퐁⃗⃗⃗⃗(⃗⃗⃗퐬⃗⃗) . The complete Frenet-Serret formulas are:

⃗퐝퐓⃗⃗ = κ 퐍⃗⃗ . 퐝퐬 ⃗퐝퐍⃗⃗ = – κ 퐓⃗⃗ + τ 퐁⃗⃗ . 퐝퐬 ⃗퐝퐁⃗⃗ = – τ 퐍⃗⃗ . 퐝퐬

Note 3: In differential geometry, the Frenet-Serret frame serves as a local intrinsic coordinate system valued for writing local laws while deemphasizing the external x-y-z space coordinates. Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright  2020, American Society for Engineering Education 8

In the case of multi-variable calculus, where the position of a particle moving along a curve described with time as a parameter, 퐑⃗⃗⃗⃗(⃗⃗퐭⃗⃗) , the velocity is, as expected, the vector derivative of position with respect to time and acceleration is the vector second derivative with respect to time.

Conclusion

While differential geometry and differential calculus have a lot in common, the engineer and/or technician should be aware of the fact that there are differences. The similarities include the graphs of the functions y = h(t) and the interpretation of the derivative as the tilt of the tangent line at each point on the curve.

One important difference is the measurement of tilt as either the angle with the horizontal axis in the case of differential geometry or as the rate of change in the case of differential calculus. A second difference is, in the case of differential geometry, the measurement of curvature is the dα rate of change of angle with respect to arc length, and in the case of differential calculus the ds d2y measurement of curvature as the second derivative of h with respect to t, . dt2 An engineer designing robots or drones must maintain the separation between the geometric and physical aspects of the problem at hand and employ the appropriate computation for each part of the problem.

References

1. Dirk J. Struik, Lectures on Classical Differential Geometry, Dover Publications, Inc., New York 2. Grossfield, A. (2018). Tilted Planes and Curvature in Three Dimensional Space Paper presented at the CIEC Annual Conference.

ANDREW GROSSFIELD Throughout his career Dr. Grossfield has combined an interest in engineering design and mathematics. He earned his BEE at CCNY. Seeing the differences between the mathematics memorized in schools and the math understood and needed by engineers has led him to a career presenting alternative mathematical insights and concepts. He was licensed in NYS as a Professional Engineer and belongs to the MAA, the ASEE and the IEEE.

Proceedings of the 2020 ASEE Gulf-Southwest Annual Conference University of New Mexico, Albuquerque Copyright  2020, American Society for Engineering Education