1. Chapter 10-Vector Valued Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

3. 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) = r 1 (t)i+r 2 (t)j+r 3 (t)k converges to the vector L = 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

4. 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) = r 1 (t)i + r 2 (t)j + r 3 (t)k be a vector-valued function. Then, if and only if Limits of Vector-Valued Functions

5. 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

6. 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.

7. 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

8. 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

9. 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) = r 1 (t)i + r 2 (t)j + r 3 (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

10. 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) = e 2t 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) = t 2 i − (1/t 2 ) 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).

11. 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) = f 1 (t) i + f 2 (t) j + f 3 (t) k is continuous then we may consider the antiderivative Antidifferentiation

12. 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) = 3t 2 i−4tj+8k that satisfies the equation F(1) = 2i−3j+2k. Antidifferentiation

13. 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 (  ) =, f’(  ) =, g (  ) =, and g’(  ) =, 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) = and F(0) = ? Quick Quiz

14. 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 Instantaneous Velocity: Instantaneous Speed:

15. 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.

16. 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).

17. 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 P 0 be a point on a space curve C. Suppose that r is a parameterization of C with P 0 = r (t 0 ). If r’(t 0 ) exists and is not the zero vector, then the tangent line to C at the point P 0 = r(t 0 ) is the line through P 0 that is parallel to vector r’(t 0 ). The tangent line is parameterized by u  r(t 0 ) + u r’(t 0 ). The Tangent Line to a Curve in Space

18. 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) =. What are parametric equations for the tangent line to C at the point P 0 = (0,  /2,  /2)? The Tangent Line to a Curve in Space

19. 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

20. 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 the acceleration 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.

21. 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) =, what is r’(2)? 2. Find symmetric equations for the tangent line to r (t) = 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) =. What is its acceleration when t = 0? Quick Quiz

22. 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 DEFINITION: Suppose that r (t) = r 1 (t) i + r 2 (t) j + r 3 (t) k is continuous for a ≤t ≤ b. We say that r is a smooth parameterization of the curve it defines if (i) the scalar-valued functions r 1, r 2, r 3 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, x 1 ], [x 1, x 2 ],..., [x N−1, b] such that the restriction of r to each subinterval is smooth, then we say that r is piecewise smooth.

23. 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 P 0 = r(0) and P 1 = r(ln (2))?

24. 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

25. 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

26. 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.

27. 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.

28. 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.

29. 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. Then for 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.

30. 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 length parameterizations of the curves that they define? EXAMPLE: Reparameterize the curve r (t) =, 0 ≤ t ≤ 2  so that it is parameterized with respect to arc length.

31. 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).

32. 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

33. 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 Quiz 1. For r(t) = t 3 i + t 2 j + t 6 k, 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+e t j+ ½ e 2t 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 unit normal vector, then what is the binormal vector?

34. Chapter 10-Vector Valued Functions 10.4 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).

35. 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.

36. 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

37. 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) = e t i + e −t j + tk. Find the curvature  r (t) at point r(t). Calculating Curvature Without Reparameterizing

38. 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

39. 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) = t 2 i + 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

40. 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 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

41. 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

42. 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 acceleration is 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 perpendicular to the osculating plane of C at r (3)? Quick Quiz

43. 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: 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

44. 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) a T (t) = a(t) · T(t) (ii) a N (t) = a(t) ·N(t) (iii) ||a(t) || 2 = (a T (t)) 2 + (a N (t)) 2. EXAMPLE: Let r(t) = sin (t) i − cos (t) j − (t 2 /2)k. Calculate a T (t) and a N (t). Express a(t) as a linear combination of T(t) and N(t).

45. 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

46. 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.

47. 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.

48. 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?

49. 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.

50. 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 points P = (x, y) for which What are the lengths of the major and minor axes? Where are the foci located?

51. 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