Chapter 10-Vector Valued Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley &...

Chapter 10-Vector Valued Functions

Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 10-Vector Valued Functions10.1 Vector-Valued Functions-

Limits, Derivatives, and Continuity


EXAMPLE: Describe the curve that is parameterized by the vector-valued function r(t) = (5 − t) i+(1+2t)j−3tk.


10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity


DEFINITION: We say that r(t) = r1(t)i+r2(t)j+r3(t)k converges to the vector L = <L1,L2,L3> as t tends to c if, for any > 0, there is a > 0 such that

If r(t) converges to L as t tends to c, then we write

and we say that L is the limit of r(t) as t tends to c.

Limits of Vector-Valued Functions




THEOREM: Let r(t) = r1(t)i + r2(t)j + r3(t)k be a vector-valued function. Then,

if and only if





EXAMPLE: Let

What is


EXAMPLE: Calculate




DEFINITION: We say that a vector-valued function r is continuous at a point c of its domain if

If r is not continuous at a point c in its domain, then we say that r is discontinuous at c.

Continuity

THEOREM: Suppose that c belongs to the domain of a vector-valued function r. Then r is continuous at c if and only if each component function of r is continuous at c.




Continuity




DEFINITION: Suppose that r is a vector-valued function that is defined on an open interval that contains c. If the limit

exists, then we call this limit the derivative of the function r at the point c and we denote this quantity by the symbols

Derivatives of Vector-Valued Functions




THEOREM: vector-valued function r (t) = r1(t)i + r2(t)j + r3(t)k is differentiable at t = c if and only if each component function of r is differentiable at c. In this case





EXAMPLE: Let r(t) = e2t i + |t| j − cos (t) k. For what values of t is r differentiable? What is r’(t) at these values?


EXAMPLE: Let f (t) = cos (t) j − ln (t) k and g(t) = t2 i − (1/t2) j + t k. Calculate (f · g) ’ (t) .

EXAMPLE: Let f (t) = tan (t) i − cos (t) k and g(t) = sin (t) j. Calculate (f × g)’ (t) .




If f (t) = f1(t) i + f2(t) j + f3(t) k is continuous then we may consider the antiderivative

Antidifferentiation




EXAMPLE: Find the antiderivative F(t) of f (t) = 3t2i−4tj+8k that satisfies the equation F(1) = 2i−3j+2k.

Antidifferentiation




1. What planar curve is described by r (t) = cos (t) i + sin (t) j, 0 ≤ t < 2?

2. If f () = <2, 0, 3>, f’() = <2, 3, 0>, g () = <0, 1, 4>, and g’() = <−1, 0, 1>, then what is (f · g)’()?

3. Referring to f and g of the preceding question, what is (f × g)’()?

4. What vector-valued function F satisfies F’(t) = <3, sin (t) , 2 exp (t)> and F(0) = <5, 5, 5>?

Quick Quiz

Chapter 10-Vector Valued Functions10.2 Velocity and Acceleration


Instantaneous Velocity:

Instantaneous Speed:


10.2 Velocity and Acceleration


DEFINITION: Suppose that a particle moving through space has position vector r(t). If r is differentiable at t, then the instantaneous velocity of the particle at time t is the vector v (t) = r’(t). We refer to v (t) = r’(t) as the velocity vector for short. The nonnegative number v(t) = is the instantaneous speed of the particle. If the instantaneous speed is

positive, then the direction vector

said to be the instantaneous direction of motion of the particle.




EXAMPLE: Let the motion of a particle in space be given by r(t) = cos (t) i + sin (t) j + t k. Sketch the curve of motion on a set of axes. Calculate the velocity vector for any t. What value does the velocity have at time t = /2? What is the speed at this time? Add the velocity vector v (/2) to your sketch, representing it by a directed line segment whose initial point is the terminal point of r(t).




DEFINTION: Let P0 be a point on a space curve C. Suppose that r is a parameterization of C with P0 = r (t0). If r’(t0) exists and is not the zero vector, then the tangent line to C at the point P0 = r(t0) is the line through P0 that is parallel to vector r’(t0). The tangent line is parameterized by u r(t0) + u r’(t0).

The Tangent Line to a Curve in Space




EXAMPLE: Consider the curve C defined by the vector-valued function

r(t) = <t cos (t) , t sin (t) , t>.

What are parametric equations for the tangent line to C at the point P0 = (0, /2, /2)?

The Tangent Line to a Curve in Space




DEFINITION: If r(t) is the position vector of a body moving through space with velocity v (t) = r’(t), then the instantaneous acceleration of the body at time t is a(t) = v’(t), provided that this derivative exists. Equivalently, we may define a(t) = r’’(t), provided that this second derivative exists.

Acceleration




EXAMPLE: Let r(t) = cos (t) i + sin (t) j + tk. Calculate the acceleration a (t). Evaluate theacceleration at t = 0, /2, 3/4, , and 2.

Acceleration

EXAMPLE: Show that the acceleration vector of a particle moving through space is always perpendicular to the velocity vector if and only if the particle travels at constant speed.




1. If r (t) = <t, t2, t3>, what is r’(2)?

2. Find symmetric equations for the tangent line to r (t) =<t, 2t2, 2 + t2> at t = 1.

3. A particle’s position is given by What is its velocity when t = 0?

4. A particle’s position is given by r (t) =< e−t, cos (t) , t2>. What is its acceleration when t = 0?

Quick Quiz

Chapter 10-Vector Valued Functions10.3 Tangent Vectors and Arc

Length


Unit Tangent VectorsDEFINITION: Suppose that r (t) = r1 (t) i + r2 (t) j + r3 (t) k is continuous for a ≤t ≤ b. We say that r is a smoothparameterization of the curve it defines if

(i) the scalar-valued functions r1, r2, r3 are all twice continuously differentiable on (a, b), and(ii) r’ (t) ≠0 for every t in (a, b).

If we can divide the interval [a, b] into finitely many subintervals [a, x1], [x1, x2], . . . , [xN−1, b] such that the restriction of r to each subinterval is smooth, then we say that r is piecewise smooth.


10.3 Tangent Vectors and Arc Length


Unit Tangent Vectors

Unit Tangent Vector to C at the point r(t):

EXAMPLE: Let Calculate the unit tangent vector T(t). What are the unit tangent vectors at P0 = r(0) and P1 = r(ln (2))?




Arc Length

Approximation of length of curve:

DEFINITION: Let C be a curve with smooth parameterization t r (t), a ≤ t ≤ b. The arc length of C is




Arc Length




Reparameterization

EXAMPLE: Suppose that r is a smooth parameterization of a curve C with P = r (t). Suppose also that p = r is a reparameterization with (s) = t. Show that the tangent line to C at the point p(s) = r (t) = P will be the same whichever parameterization we use.




Reparameterization

THEOREM: Suppose that C is a curve with smooth parameterization t r (t), a ≤ t ≤ b. Let : [c, d] [a, b] be a continuously differentiable increasing function and let p = r be a reparameterization of C. Then the arc length of C when computed using the reparameterization p is equal to the arc length of C when computed using the parameterization r.




Parameterizing A Curve by Arc Length

DEFINITION: Let L be the length of a curve C. A parameterization p (s), 0 ≤ s ≤ L, of C is called the arc length parameterization of C if the arc length between p(0) and p(s) is equal to s for every s in the interval [0,L]. We also say that p parameterizes C with respect to arc length.





THEOREM: Let s p (s), 0 ≤ s ≤ L be the arc length parameterization of a curve C. Thenfor all s. Moreover, for every s, T(s) = p’(s).

Conversely, if t r (t), 0 ≤ t ≤ b is a smooth parameterization of a curve C such that for all t, then r is the arc length parameterization of C and b is the length of C.





EXAMPLE: Are r(t) = cos (2t) i − sin (2t) k, 0 ≤ t ≤ and p(u) = cos (u) i − sin (u) k, 0 ≤ u ≤ 2 arc lengthparameterizations of the curves that they define?

EXAMPLE: Reparameterize the curve

r (t) = <cos (t) , sin (t) , t>, 0 ≤ t ≤ 2

so that it is parameterized with respect to arc length.




Unit Normal Vectors

DEFINITION: If t r (t) is a smooth parameterization of a curve C with T’(t) ≠ 0, then the vector

is called the principal unit normal to C at r (t).




Unit Normal Vectors




Quick Quiz1. For r(t) = t3i + t2j + t6k, what is the unit tangent vector T(1)?

2. What integral represents the arc length of the plane curve parametrized by r(t) = t i+et j+ ½ e2t k, 0 ≤ t ≤ 1?

3. If r is an arc length parameterization of a curve of length 6, what is

4. If, at a point on a curve, is the unit tangent vector and is the principal unitnormal vector, then what is the binormal vector?

Chapter 10-Vector Valued Functions10.4 Curvature


DEFINITION: Suppose that s p (s) is a smooth arc length parameterization of a curve C. The quantity

is called the curvature of C at the point r (s) .


10.4 Curvature


EXAMPLE: Let r(s) = cos(s/) i + sin(s/) j for some positive constant . Calculate the curvature at each value of s.


10.4 Curvature


THEOREM: Suppose that r is a smooth parameterization of a curve C. Then

is the curvature of C at the point r (t). The curvature at r (t) may also be expressed as

Calculating Curvature Without Reparameterizing


10.4 Curvature


EXAMPLE: Let C be the curve parameterized by r(t) = eti + e−tj + tk. Find the curvature r (t) at point r(t).

Calculating Curvature Without Reparameterizing


10.4 Curvature


DEFINITION: Let p be a smooth arc length parameterization of a space curve C. If (s) > 0, then the osculating circle (or the circle of curvature) of C at p(s) is the unique circle satisfying the following conditions:

(a) the circle has radius (s) = 1/ (s);(b) the circle has center p(s) + (s)N(s). This point is called the center of curvature of C at p(s);(c) the circle lies in the plane determined by the vectors N(s) and T(s).

This plane is called the osculating plane of C at p(s).

Osculating Circle


10.4 Curvature


EXAMPLE: Let C be the curve parameterized by r(t) = t2i + tj + tk. Find the curvature r (t) at the point r(t). Also determine the radius of curvature, principal unit normal, and center of the osculating circle at the point r(t). What is the Cartesian equation of the osculating plane at r(t)?

Osculating Circle


10.4 Curvature


THEOREM: Suppose that x (t) and y (t) are twice differentiable functions that define a planar curve C by means of the parametric equations x = x (t), y = y (t). Then, at any point P = ((x (t) , y (t)) for which the velocity vector <x’(t), y’(t)> is not 0, the curvature of C is given by

In particular, the curvature of the graph of y = f (x) at the point (x, f (x)) is

Planar Curves


10.4 Curvature


EXAMPLE: Find the circle of curvature of the curve y = sin (2x) at the point P = (/4, 1) .

Planar Curves


10.4 Curvature


1. The position p (s) of a moving particle is such that What is the curvature of the trajectory of the particle at s = 2?

2. As a particle passes through point P, its speed is 2, the magnitude of its acceleration is 6, and its accelerationis perpendicular to its velocity. What is the curvature of the trajectory of the particle at P?

3. If t r (t) is a smooth parameterization of a curve C and if r (3) > 0, then what unit vectors are perpendicularto the osculating plane of C at r (3)?

Quick Quiz

Chapter 10-Vector Valued Functions10.5 Applications of Vector-

Valued Functions to Motion


THEOREM: Suppose that t r (t) is a smooth parameterization of a space curve C. Let v(t) = || r’(t) || denote the speed of a particle moving along the curve with position vector r(t). Then the acceleration vector a(t) = r’’(t) can be decomposed as the sum of two vectors, one with direction T(t), the unit tangent to C at r(t), and the other with direction N(t), the principal unit normal to C at r (t). The decomposition has the form

where and


10.5 Applications of Vector-Valued Functions to Motion


THEOREM: The tangential and normal components of acceleration satisfy

(i) aT (t) = a(t) · T(t)(ii) aN (t) = a(t) ·N(t)(iii) ||a(t) ||2 = (aT (t))2 + (aN (t))2.

EXAMPLE: Let r(t) = sin (t) i − cos (t) j − (t2/2)k. Calculate aT (t) and aN (t) . Express a(t) as a linear combination of T(t) and N(t).




EXAMPLE: Suppose that a > b > 0. A force F acts on a particle of mass m in such a way that the particle moves in the xy-plane with position described by r(t) = a cos (t) i + b sin (t) j. Show that F is a central force field.

Central Force Fields




Central Force Fields

THEOREM: If a particle moving in space is subject only to a central force field then the particle’s trajectory lies in a plane.

THEOREM: If a moving particle is subject only to a central force field then the particle’s position vector sweeps out a region whose area A(t) has a constant rate of change with respect to t.




Ellipses EXAMPLE: Suppose that a > b > 0. Show that the curve described by r (t) = a cos (t) i + b sin (t) j, 0 ≤t ≤ 2, is an ellipse. Where are the foci located?

DEFINITION: Let c denote the half-distance between the foci of an ellipse. Let a denote half the length of the major axis of the ellipse. The quantity e = c/a is called the eccentricity of the ellipse.




Ellipses

EXAMPLE: If the length of the major axis of an ellipse is double the length of its minor axis, then what is the eccentricity of the ellipse?




Ellipses

THEOREM: Let C be a curve in the plane. Then C is an ellipse if and only if there is a real number e with 0 < e < 1, a point F, and a line D such that C is the locus of all points P that satisfy If C is the ellipse defined by the equation then

i) The eccentricity of C is e.ii) C lies on one side of D.iii) F is a focus of C, the focus closest to D.iv) The major axis of C is perpendicular to D.




Ellipses

EXAMPLE: Suppose that F = (0, 0) and D is the line x = 2. What is the Cartesian equation for the locus of pointsP = (x, y) for which

What are the lengths of the major and minor axes? Where are the foci located?




Quick Quiz

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Chapter 10-Vector Valued Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley &...

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