Vector Functions & Space Curves MATH 2110Q

Transcription

1 Vector Functions & Space Curves Vector Functions & Space Curves

2 Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors as outputs. Before Functions take a number as input and assign a number as output. input x f(x) output Example: f(3) = 11 Now Vector functions take a number as input and assign a vector as output. input t r(t) output Example: r(1.2) = 2i j Vector Functions & Space Curves

3 Vector Functions Example Because we re working (primarily) in R 3, our vector functions look like this: r(t) = f(t), g(t), h(t) = f(t)i + g(t)j + h(t)k component functions Example: v(t) = t, 2t + 1, sin(7t) Vector Functions & Space Curves

4 Vector Functions Graphing We already saw the example of lines. The vector function r(t) = r 0 + tv describes a straight line through 3D space. Vector Functions & Space Curves

5 Vector Functions Graphing More generally, graphing a continuous (?) vector function produces a space curve, i.e. a curve through 3D space. Vector Functions & Space Curves

6 Vector Functions Parametric Equations We can split up the vector function r(t) = f(t), g(t), h(t) into its components to get a system of parametric equations: x = f(t), y = g(t), z = h(t). We will sometimes call t the parameter. When we use vector or parametric equations to draw a space curve C, we think t = time and the curve is traced out by a moving particle whose position is given by the functions. Vector Functions & Space Curves

7 Example 1 Problem Sketch the curve whose vector equation is The parametric equations are r(t) = cos t, sin t, t. x = cos t, y = sin t, z = t. This is complicated. Can we simplify it? Can we figure out just part of the problem first? Do we recognize any patterns? Vector Functions & Space Curves

8 Example 1 Problem Sketch the curve whose vector equation is r(t) = cos t, sin t, t. What s going on in the xy-plane? We saw this in Calculus II. x = cos t, y = sin t. Vector Functions & Space Curves

9 Example 1 Problem Sketch the curve whose vector equation is r(t) = cos t, sin t, t. So x = cos t and y = sin t loop around and around the unit circle. Meanwhile, what does z = t do? Vector Functions & Space Curves

10 Example 1 Problem Sketch the curve whose vector equation is r(t) = cos t, sin t, t. Vector Functions & Space Curves

11 Example 2 Problem Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2. What is this problem asking for? Vector Functions & Space Curves

12 Example 2 Problem Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2. Vector Functions & Space Curves

13 Example 2 Problem Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2. Vector Functions & Space Curves

14 Example 2 Problem Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2. Just like before, the fact that we re making loops inside a cylinder of radius 1 means x and y are taken care of. We can set What about z? x = cos t, y = sin t. Vector Functions & Space Curves

15 Example 2 Problem Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2. Just like before, the fact that we re making loops inside a cylinder of radius 1 means x and y are taken care of. We can set What about z? x = cos t, y = sin t. Because y + z = 2, z = 2 y = 2 sin t. So r(t) = cos t, sin t, 2 sin t. Vector Functions & Space Curves

16 Exercise Find a vector function that represents the curve of intersection of the paraboloid z = 4x 2 + y 2 and the parabolic cylinder y = x 2. Vector Functions & Space Curves

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18 Limits of Vector Functions If r(t) = f(t), g(t), h(t), then lim r(t) = lim f(t), lim g(t), lim h(t) t a t a t a t a provided the limits of the component functions exist.

19 Limits of Vector Functions Example Find the limit. ) lim (e 3t i + t2 t 0 sin 2 j + cos 2tk t lim t 0 e 3t = e 0 = 1. lim t 0 cos 2t = cos 0 = 1. lim t 0 t 2 sin 2 t =?

20 Limits of Vector Functions Example Find the limit. ) lim (e 3t i + t2 t 0 sin 2 j + cos 2tk t lim t 0 e 3t = e 0 = 1. lim t 0 cos 2t = cos 0 = 1. lim t 0 t 2 sin 2 t = 1.

21 Limits of Vector Functions Example Find the limit. ) lim (e 3t i + t2 t 0 sin 2 j + cos 2tk t ) lim (e 3t i + t2 t 0 sin 2 j + cos 2tk = i + j + k. t

22 Derivatives of Vector Functions dr dt = r(t + h) r(t) r (t) = lim h 0 h

23 Derivatives of Vector Functions From the pictures, we see that r (t) is tangent to the curve described by r(t). Sometimes we want a unit tangent vector, which we write T(t). T(t) = r (t) r (t).

24 Derivatives of Vector Functions Theorem If r(t) = f(t), g(t), h(t) and f, g, h are differentiable, then r (t) = f (t), g (t), h (t).

25 Derivatives of Vector Functions Theorem If r(t) = f(t), g(t), h(t) and f, g, h are differentiable, then r (t) = f (t), g (t), h (t). r r (t) = lim t 0 t = lim 1 f, g, h t 0 t f = lim t 0 t, g t, h t f = lim t 0 t, lim g t 0 t, lim h t 0 t = f (t), g (t), h (t).

26 Derivatives of Vector Functions Example (a) Find the derivative of r(t) = (1 + t 3 )i + te t j + sin 2tk.

27 Derivatives of Vector Functions Example (a) Find the derivative of r(t) = (1 + t 3 )i + te t j + sin 2tk. d dt (1 + t3 ) =

28 Derivatives of Vector Functions Example (a) Find the derivative of r(t) = (1 + t 3 )i + te t j + sin 2tk. d dt (1 + t3 ) = 3t 2 d ( te t ) = dt

29 Derivatives of Vector Functions Example (a) Find the derivative of r(t) = (1 + t 3 )i + te t j + sin 2tk. d dt (1 + t3 ) = 3t 2 d ( te t ) = e t te t = (1 t)e t dt d sin 2t = dt

30 Derivatives of Vector Functions Example (a) Find the derivative of r(t) = (1 + t 3 )i + te t j + sin 2tk. d dt (1 + t3 ) = 3t 2 d ( te t ) = e t te t = (1 t)e t dt d sin 2t = 2 cos 2t dt r (t) = 3t 2 i + (1 t)e t j + 2 cos 2tk

31 Derivatives of Vector Functions Example (b) Find the unit tangent vector of r(t) = (1 + t 3 )i + te t j + sin 2tk at t = 0.

32 Derivatives of Vector Functions Example (b) Find the unit tangent vector of at t = 0. r(t) = (1 + t 3 )i + te t j + sin 2tk r (t) = 3t 2 i + (1 t)e t j + 2 cos 2tk So r (0) = j + 2k and r (0) = 5. Thus T(0) = r (0) r (0) = 1 5 j k

33 Derivatives of Vector Functions Facts

34 Derivatives of Vector Functions Facts Note: r (t), r (t), etc. work the same as in Calculus I.

35 Summary Limits of vector functions work componentwise, i.e. piece by piece. Derivatives of vector functions work componentwise. How do integrals work?

36 Integrals of Vector Functions You guessed it. b a ( b r(t) dt = b a a r(t) dt = ) ( b ) ( b ) f(t) dt i + g(t) dt j + h(t) dt k a a b b b a f(t) dt, a g(t) dt, a h(t) dt

37 Integrals of Vector Functions Example If r(t) = 2 cos t, sin t, 2t, then r(t) dt = 2 cos t dt, sin t dt, 2t dt where C is a vector constant. = 2 sin t, cos t, t 2 + C,

38 Review

39 Review basic properties of integrals change of variables (u-substitution) integration by parts