Chapter 14: Vector Calculus

Transcription

1 Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity and Differentiability e. Properties f. Integration g. Properties of the Integral Section 14.2 The Rules of Differentiation a. Combining Vector Functions b. Differentiation Rules c. Differentiation Rules, Leibniz s Notation d. Properties Section 14.3 Curves a. Differentiable Curves b. Parametrized Curves c. Tangent Vector d. Direction of Tangent Vector e. Tangent Line f. Intersecting Curves g. Unit Tangent Vector; Principal Normal Vector h. Reversing the Direction of a Curve i. Spiraling Helix Section 14.4 Arc Length a. Definition: Arc Length b. Arc Length Formula c. Parametrizing a Curve d. Tangent Vector Properties Section 14.5 Curvilinear Motion; Curvature a. Vector Viewpoint b. Curvature c. Plane Curves d. Circles e. Space Curves f. Components of Acceleration g. More Properties Section 14.6 Vector Calculus in Motion Newton s Second Law Introduction to Vector Mechanics Momentum Angular Momentum Torque Central Force Initial-Value Problems Section 14.7 Planetary Motion Newton s Second Law; Motion for Extended Three Dimensional Objects Kepler s Law

2 Vector Calculus Functions such as f (t) = 2 + 3t, f (t) = at 2 + bt + c, f (t) = sin 2t assign real numbers to real numbers. They are called real-valued functions of a real variable, for short, scalar functions. Functions such as f (t) = r + td, f (t) = t 2 a + t b + c, f (t) = sin t a + cos t b 0 assign vectors to real numbers. They are called vector-valued functions of a real variable, for short, vector functions. Vector functions can be built up from scalar functions in an obvious manner. From scalar functions f 1, f 2, f 3 that share a common domain we can construct the vector function f (t) = f 1 (t) i + f 2 (t) j + f 3 (t) k. The functions f 1, f 2, f 3 are called the components of f. A number t is in the domain of f iff it is in the domain of each of its components.

3 Limit, Continuity, Vector Derivative Remark The converse of (14.1.2) is false. You can see this by setting f (t) = k and taking L = k.

4 Limit, Continuity, Vector Derivative

5 Limit, Continuity, Vector Derivative The limit process can be carried out component by component: then let f (t) = f 1 (t) i + f 2 (t) j + f 3 (t) k and let L = L 1 i + L 2 j + L 3 k;

6 Limit, Continuity, Vector Derivative Continuity and Differentiability As you would expect, f is said to be continuous at t 0 provided that 0 ( ) = f( ) lim f t t t t Thus, by (14.1.4), f is continuous at t 0 iff each component of f is continuous at t 0. To differentiate f, we form the vector (1/h) [f (t + h) f (t)] and write it as f 0 ( t+ h) f( t) h

7 Limit, Continuity, Vector Derivative

8 Limit, Continuity, Vector Derivative Integration Just as we can differentiate component by component, we can integrate component by component. For f (t) = f 1 (t) i + f 2 (t) j + f 3 (t) k continuous on [a, b], we set

9 Limit, Continuity, Vector Derivative Properties of the Integral

10 The Rules of Differentiation Vector functions with a common domain can be combined in many ways to form new functions. From f and g we can form the sum f + g: (f + g)(t) = f (t) + g(t). We can form scalar multiples αf and thus linear combinations αf + βg: We can form the dot product f g: (αf )(t) = αf (t), (αf + βg)(t) = αf (t) + βg(t). We can also form the cross product f g: (f g)(t) = f (t) g(t). (f g)(t) = f (t) g(t). There are two ways of bringing scalar functions (real-valued functions) into this mix. If a scalar function u shares a common domain with f, we can form the product uf: (uf )(t) = u(t) f (t). If u(t) is in the domain of f for each t in some interval, then we can form the composition f u: (f u)(t) = f(u(t)).

11 The Rules of Differentiation

12 The Rules of Differentiation

13 The Rules of Differentiation

14 Curves A linear function r(t) = r 0 + t d, d 0 traces out a line, and it does so in a particular direction, the direction imparted to it by increasing t. More generally, a differentiable vector function r(t) = x(t) i + y(t) j + z(t) k traces out a curved path, and it does so in a particular direction, the direction imparted to it by increasing t. The directed path C (called by some the oriented path) traced out by a differentiable vector function is called a differentiable parametrized curve.

15 Curves We draw a distinction between the parametrized curve and the parametrized curve C 1 : r 1 (t) = cos t i + sin t j, t [0, 2π] C 2 : r 2 (u) = cos (2π u) i + sin (2π u) j, u [0, 2π]. The first curve is the unit circle traversed counterclockwise; the second curve is the unit circle traversed clockwise.

16 Curves Tangent Vector, Tangent Line

17 Curves

18 Curves If r (t 0 ) 0, then r (t 0 ) is tangent to the curve at the tip of r(t 0 ). The tangent line at this point can be parametrized by setting

19 Curves Intersecting Curves Two curves C 1 : r 1 (t) = x 1 (t) i + y 1 (t) j + z 1 (t) k, intersect iff there are numbers t and u for which r 1 (t) = r 2 (u). C 2 : r 2 (u) = x 2 (u) i + y 2 (u) j + z 2 (u) k The angle between C 1 and C 2 at a point where r 1 (t) = r 2 (u) is by definition the angle between the corresponding tangent vectors r' 1 (t) and r' 2 (u).

20 Curves The Unit Tangent, the Principal Normal, the Osculating Plane Suppose now that the curve C : r(t) = x(t) i + y(t) j + z(t) k is twice differentiable and r'(t) is never zero. Then at each point y(t), z(t)) of the curve, there is a unit tangent vector: P(x(t), If the unit tangent vector is not changing in direction (as in the case of a straight line), then T (t) = 0. If T (t) 0, then we can form what is called the principal normal vector:

21 Curves Reversing the Direction of a Curve We make a distinction between the curve and the curve r = r(t), t [a, b] R(u) = r(a + b u), u [a, b]. Both vector functions trace out the same set of points, but the order has been reversed. Whereas the first curve starts at r(a) and ends at r(b), the second curve starts at r(b) and ends at r(a): R(a) = r(a + b a) = r(b), R(b) = r(a + b b) = r(a).

22 Curves The function r(t) = a cos t i + a sin t j + bt k, t [0, 2π] traces out one turn of a spiraling helix (Figure ), the direction of transversal indicated by the little arrows. The function R(u) = a cos (2 u)π i + a sin (2 u)π j + b(2 u)π k, u [0, 2] produces the same path but in the opposite direction. (Figure )

23 Arc Length

24 Arc Length

25 Arc Length Parametrizing a Curve by Arc Length Suppose that C : r = r(t), t [a, b] is a continuously differentiable curve of length L with nonzero tangent vector r (t). The length of C from r(a) to r(t) is t ( ) ( ) s t = r u du a Since ds/dt = r (t) > 0, the function s = s(t) is a one-to-one increasing function. Thus, no two points of C can lie at the same arc distance from r(a). It follows that for each s [0, L], there is a unique point R(s) on C at arc distance s from r(a).

26 Arc Length

27 Curvilinear Motion; Curvature Curvilinear Motion from a Vector Viewpoint We can describe the position of a moving object at time t by a radius vector r(t). As t ranges over a time interval I, the object traces out some path C : r(t) = x(t) i + y(t) j + z(t) k, t I. If r is twice differentiable, we can form r (t) and r (t). In this context these vectors have special names and special significance: r (t) is called the velocity of the object at time t, and r (t) is called the acceleration.

28 Curvilinear Motion; Curvature Curvature Let C : r = r(t), t I be a twice differentiable curve with nonzero tangent vector r (t). At each point the curve has a unit tangent vector T. While T cannot change in length, it can change in direction. At each point of the curve the change in direction of T per unit of arc length is given by the derivative dt/ds. The magnitude of this change in direction per unit of arc length, the number is called the curvature of the curve. As you would expect, κ = dt ds

29 Curvilinear Motion; Curvature The Curvature of a Plane Curve Figure shows a plane curve. At a point P we have attached the unit tangent vector T and drawn the tangent line. The angle marked φ is the inclination of the tangent line measured in radians. As P moves along the curve, angle φ changes. The curvature at P can be interpreted as the magnitude of the change in φ per unit of arc length. φ

30 Curvilinear Motion; Curvature

31 Curvilinear Motion; Curvature Calculating the Curvature of a Space Curve Example Calculate the curvature of the circular helix r(t) = a sin t i + a cos t j + t k. (a > 0) Solution We will use the Leibniz notation. Here Therefore dr ds dr = a ti a tj+ k = = a + dt dt dt dr dt acost i asint j+ k T = = dr dt 2 a + 1 dt asinti acostj dt a = and = dt 2 2 a + 1 dt a cos sin, 1 dt dt a 2 a + 1 a 2 2 κ = = = ds dt a + 1 a + 1 The circular helix is a curve of constant curvature.

32 Curvilinear Motion; Curvature Components of Acceleration If we take the dot product of T with a, we get Therefore, T a = a T (T T) + a N (T N) = a T.

33 Curvilinear Motion; Curvature Crossing T with a, we get T a = a T (T T) + a N (T N) = a N (T N) and so T a = a N T N = a N T N sin (π/2) = a N. Therefore Since a N = κ(ds/dt) 2, it follows that

34 Vector Calculus In Mechanics The tools we have developed in the preceding sections have their premier application in Newtonian mechanics, the study of bodies in motion subject to Newton s laws. The heart of Newton s mechanics is his second law of motion: force = mass acceleration. We have worked with Newton s second law, but only in a very restricted context: motion along a coordinate line under the influence of a force directed along that same line. In that special setting, Newton s law was written as a scalar equation: F = ma. In general, objects do not move along straight lines (they move along curved paths) and the forces on them vary in direction. What happens to Newton s second law then? It becomes the vector equation F = ma. This is Newton s second law in its full glory.

35 Vector Calculus In Mechanics An Introduction to Vector Mechanics We are now ready to work with Newton s second law of motion in its vector form: F = ma. Since at each time t we have a(t) = r''(t), Newton s law can be written This is a second-order differential equation.

36 Vector Calculus In Mechanics Momentum We start with the idea of momentum. The momentum p of an object is the mass of the object times the velocity of the object: p = mv. To indicate the time dependence we write Assume that the mass of the object is constant. Then differentiation gives p (t) = mr (t) = F (t). Thus, the time derivative of the momentum of an object is the net force on the object. If the net force on an object is continually zero, the momentum p(t) is constant. This is the law of conservation of momentum:

37 Vector Calculus In Mechanics Angular Momentum If the position of the object at time t is given by the radius vector r(t), then the object s angular momentum about the origin is defined by the formula The angular momentum comes entirely from the component of velocity that is perpendicular to the radius vector.

38 Vector Calculus In Mechanics Torque How the angular momentum of an object changes in time depends on the force acting on the object and on the position of the object relative to the origin that we are using.

39 Vector Calculus In Mechanics A force F = F(t) is called a central force (radial force) if F(t) is always parallel to r(t). (Gravitational force, for example, is a central force.) For a central force, the cross product r(t) F(t) is always zero. Thus a central force produces no torque about the origin. As you will see, this places severe restrictions on the kind of motion possible under a central force.

40 Vector Calculus In Mechanics Initial-Value Problems In physics one tries to make predictions about the future on the basis of current information and a knowledge of the forces at work. In the case of an object in motion, the task can be to determine r(t) for all t given the force and some initial conditions. Frequently the initial conditions give the position and velocity of the object at some time t 0. The problem then is to solve the differential equation F= m r subject to conditions of the form r(t 0 ) = r 0, v(t 0 ) = v 0. Such problems are known as initial-value problems.

41 Planetary Motion Newton s Second Law of Motion for Extended Three-Dimensional Objects The total external force on an extended three-dimensional object is thus the total mass of the object times the acceleration of the center of mass.

42 Planetary Motion A Derivation of Kepler s Laws from Newton s Laws of Motion and His Law of Gravitation The gravitational force exerted by the sun on a planet can be written in vector form as mm F r = G r. r ( ) ( ) 3