chapter 10-vector valued functions calculus, 2ed, by blank & krantz, copyright 2011 by john...
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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
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Describe the curve that is parameterized by the vector-valued function r(t) = (5 − t) i+(1+2t)j−3tk.
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
THEOREM: Let r(t) = r1(t)i + r2(t)j + r3(t)k be a vector-valued function. Then,
if and only if
Limits of Vector-Valued Functions
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Let
What is
Limits of Vector-Valued Functions
EXAMPLE: Calculate
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Continuity
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Derivatives of Vector-Valued Functions
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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?
Derivatives of Vector-Valued Functions
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) .
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
If f (t) = f1(t) i + f2(t) j + f3(t) k is continuous then we may consider the antiderivative
Antidifferentiation
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Find the antiderivative F(t) of f (t) = 3t2i−4tj+8k that satisfies the equation F(1) = 2i−3j+2k.
Antidifferentiation
Chapter 10-Vector Valued Functions
10.1 Vector-Valued Functions-Limits, Derivatives, and Continuity
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Instantaneous Velocity:
Instantaneous Speed:
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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).
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.2 Velocity and Acceleration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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))?
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Arc Length
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parameterizing A Curve by 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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parameterizing A Curve by Arc Length
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.
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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).
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Normal Vectors
Chapter 10-Vector Valued Functions
10.3 Tangent Vectors and Arc Length
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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) .
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Let r(s) = cos(s/) i + sin(s/) j for some positive constant . Calculate the curvature at each value of s.
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Find the circle of curvature of the curve y = sin (2x) at the point P = (/4, 1) .
Planar Curves
Chapter 10-Vector Valued Functions
10.4 Curvature
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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).
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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?
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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.
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
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?
Chapter 10-Vector Valued Functions
10.5 Applications of Vector-Valued Functions to Motion
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz