Chapter 14: Vector Calculus

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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

Chapter 14: Vector Calculus

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

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Vector Calculus

Functions such as

f (t) = 2 + 3t, f (t) = at² + 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) = t² a + t b + c, f (t) = sin t a + cos t b

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 f₁, f₂, f₃ that share a common domain we can construct the vector function

f (t) = f₁(t) i + f₂(t) j + f₃(t) k.

The functions f₁, f₂, f₃ 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.

0

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Remark The converse of (14.1.2) is false. You can see this by setting f (t) = k and taking L = −k.

Limit, Continuity, Vector Derivative

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Limit, Continuity, Vector Derivative

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The limit process can be carried out component by component:

let f (t) = f₁(t) i + f₂(t) j + f₃(t) k and let L = L₁ i + L₂ j + L₃ k;

then

Limit, Continuity, Vector Derivative

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Continuity and Differentiability

As you would expect, f is said to be continuous at t₀ provided that

Thus, by (14.1.4), f is continuous at t₀ iff each component of f is continuous at t₀. To differentiate f, we form the vector (1/h) [f (t + h) − f (t)] and write it as

( ) ( )

0

lim 0

t t t t

→ f =f

(

^t ^h

) ( )

^t

h + −

f f

Limit, Continuity, Vector Derivative

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Limit, Continuity, Vector Derivative

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Integration

Just as we can differentiate component by component, we can integrate component by component. For f (t) = f₁(t) i + f₂(t) j + f₃(t) k continuous on [a, b], we set

Limit, Continuity, Vector Derivative

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Properties of the Integral

Limit, Continuity, Vector Derivative

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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:

(αf )(t) = αf (t), (αf + βg)(t) = αf (t) + βg(t).

We can form the dot product f · g:

(f · g)(t) = f (t) · g(t).

We can also form the cross product f × g:

(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)).

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The Rules of Differentiation

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The Rules of Differentiation

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The Rules of Differentiation

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Curves

A linear function

r(t) = r₀ + 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.

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Curves

We draw a distinction between the parametrized curve

C₁ : r₁(t) = cos t i + sin t j, t ∈ [0, 2π]

and the parametrized curve

C₂ : r₂(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.

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Curves

Tangent Vector, Tangent Line

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Curves

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Curves

If r´(t₀) ≠ 0, then r´(t₀) is tangent to the curve at the tip of r(t₀). The tangent line at this point can be parametrized by setting

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Curves

Intersecting Curves Two curves

C₁ : r₁(t) = x₁(t) i + y₁(t) j + z₁(t) k, C₂ : r₂(u) = x₂(u) i + y₂(u) j + z₂(u) k intersect iff there are numbers t and u for which

r₁(t) = r₂(u).

The angle between C₁ and C₂ at a point where r₁(t) = r₂(u) is by definition the angle between the corresponding tangent vectors r'₁(t) and r'₂(u).

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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 P(x(t), y(t), z(t)) of the curve, there is a unit tangent vector:

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:

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Curves

Reversing the Direction of a Curve We make a distinction between the curve

r = r(t), t ∈ [a, b]

and the curve

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).

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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 14.3.13), 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 14.3.14)

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Arc Length

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Arc Length

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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

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).

( )

^t

( )

s t = ∫

a

r u ′ du

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Arc Length

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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.

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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,

d κ = dsT

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Curvilinear Motion; Curvature

φ

The Curvature of a Plane Curve

Figure 14.5.3 shows a plane curve. At a point P we have attached the unit tangent vector T and drawn the tangent line. The angle marked is the inclination of the tangent line measured in radians. As P moves along the curve, angle changes. The curvature at P can be interpreted as the

magnitude of the change in per unit of arc length.

φ

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Curvilinear Motion; Curvature

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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

2

2 2

cos sin , 1

cos sin 1 sin cos

and

1 1

d ds d

a t a t a

dt dt dt

d dt a t a t

d dt a

d a t a t d a

dt a dt a

= − + = = +

− +

= =

+

− −

= =

+ +

r r

i j k

r i j k

T r

T i j T

The circular helix is a curve of constant curvature.

2 2 2

1 1 1

dT dt a a a

ds dt a a

κ = = ⁺ =

+ + Therefore

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Curvilinear Motion; Curvature

Components of Acceleration

If we take the dot product of T with a, we get

T· a = a_T(T· T) + a_N(T·N) = a_T. Therefore,

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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

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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.

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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.

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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:

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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.

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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.

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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.

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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 t₀. The problem then is to solve the differential equation

subject to conditions of the form

r(t₀) = r₀, v(t₀) = v₀. Such problems are known as initial-value problems.

= m ′′

F r

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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.

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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

(∗)

( )

^G ^mM₃

= − r

F r r.