CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS

Transcription

1 CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS 4.1 Vector Functions 4.2 Calculus of Vector Functions 4.3 Tangents

2 REVIEW: Vectors Scalar a quantity only with its magnitude Example: temperature, speed, mass, volume Vector a quantity with its magnitude and its direction Example: velocity acceleration, force z k i j y x Vector is denoted by A, A or A. OA - position vector AB - displacement vector

3 Vector Form of a Line Segment If r 0 is vector in 2-space or 3-space with its initial point at the origin, then the line that passes through the terminal point of r 0 and is parallel to the vector. v can be expressed in vector form as r = r 0 + tv r = r + t( r r ) or r = (1 r + tr Vector Algebra Addition associative law A + ( B + C) = ( A + B) + C commutative law A + B = B + A

4 multiplication by scalar B kb distributive law kb ( + C) = kb) + kc unit vectors: i, j, k A =< A, A, A >= Ai + A j + A k i=< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 > Unit vector, u of v: v u = v

5 scalar product (dot produc A i B = A B cos θ AB A i B = B i A A i( B + C ) = A i B + A i C A i B = AB + AB + AB vector product (cross produc A B = A B sin θ AB A B A B

6 A B = B A B A A ( B C ) ( A B ) C i j k A B = A A A B B B A A A A i ( B C) = B B B C C C 1 2 3

7 4.1 Vector Functions Vector-valued function Definition A vector-valued function ( or simply vector function) is a function whose domain is a set of real numbers and whose range is a set of vectors. Vector function, rt (): rt () =< ft (), gt (), ht () > where f, g, and h are real-valued functions called the component functions of r; t is the independent variable (time). Note If domains are intervals of real numbers, the vector functions represent a space curve If domains are regions in the plane, the vector functions represent surfaces in space Graph of a Vector Function

8 Consider a particle moving through space during a time interval I. The coordinates are seen as functions defined as: x = f(), t y = g(), t z = h(), t t I (1) The points ( xyz,,) = ((), ft gt (), ht ()) make up the curve in space, called the particle s path. Eqn. (1) parameterize the curve. A curve in space can also be represented in vector form. The vector rt () = fti () + gtj () + htk () is the particle s position vector.

9 Definition Let F be a vector function, and suppose the initial point of the vector Ft () is at the origin. The graph of F is the curve traced out by the terminal point of the vector Ft () as t varies over the domain set D.

10 4.1.3 Vector Functions Operations Theorem Let F and Ĝ be vector functions of the real variable t, and let f ( be a scalar function. Then ( ) i. F + G ( t ) = F ( t ) + G ( t ) )( ii. ( f F = f ( F ( ( ) iii. F G ( t ) = F ( t ) G ( t ) ( ) iv. F G ( t ) = F ( t ) G ( t ) Question 1 Sketch the graph of the vector function 2 rt () = ti + ( t + 3) j, 2 t 2 Label the position of r( 2), r (1) and r (2).

11 Question 2 Sketch the graph of the vector function rt () = 3 t i+ 2tj + 3t 2 k Label the position of r(0). ( ) ( ) Question 3 F () t = t 2 i + 2t j+ costk Gt () = ti + t 2 j+ 5k If ( + ) (a) F G () t (c)( ) F G () t (b) (d), find and (sin tf )( t ) F Gt () Ft ()

12 4.2 Calculus of Vector Functions Vector Derivatives Definition The derivative F df = of a vector F is defined as: F t t F t F ( + Δ ) ( ) ( = lim Δt 0 Δt where F ( = f (, g(, h( Theorem The vector function F ~ ~ ~ ( = f ( i + g( j + h( k is differentiable whenever the component functions f(, g( and h( are all differentiable. F ~ ~ ~ ( = f ( i + g ( j + h ( k

13 Example d d (3 i + sin tj ) = ti + tj + tek = 2 t (3 cos 4 ) Higher Vector Derivatives Higher derivatives of a vector function F are obtained by successively differentiating the components of F ~ ~ ~ ( = f ( i + g( j + h( k. The second derivative of F is the function F ( = [ F ] ~ ~ ( = f ( i + g ( j + h ( k and the third derivative F ( is the derivative of F ( and so forth. ~

14 Example Let F( e ~ i + e ~ j 2t 2t t ~ 2 = + te (i) unit tangent vector T (0) (ii) F (0) (iii) F ( F ( k. Find Theorem: Differentiation rules Suppose F and Ĝ are differentiable vector functions and c is a scalar and f is a real valued function. Then d ( + ( i. [ F G ] F ( G ( d ( = ii. [ cf ] = cf ( d ( iii. [ f ( F ] = f ( F + f ( F ( + (

15 d ( ( iv. [ F G ] = F ( G + F G ( d ( ( v. [ F G ] = F ( G + F G ( d ( vi. [ F f ( ] = f ( F ( f ( ), chain rule Likewise, we can obtained the partial derivatives of a multivariable vector function. Suppose ( ( ( ( Rt ( ) = f() ti+ gt () j + htk () is a differentiable functions of n variables, t 1, t 2,, t n. Then, the partial derivative of R ( t ) is f g h Rt ( ) = i + j+ k t1 t 1 t 1 t 1

16 and f g h Rt ( ) = i + j+ k t t t t n n n n Rt ( ) f g h = i+ j + k t t t t t t t t 1 m 1 m 1 m 1 m Example Let 2 2 Ruv (,) = 2 uvi + ( u 2) v j + ( u+ v ) k. Find the partial derivatives and 2 R uv. R u, R v, 2 R 2 u, 2 R 2 v Vector Integrals F ~ ~ ~ ( f ( i + g( j + h( k a t Let = where f, g and h are continous functions for b). Then,

17 t (i) the definite integral of F ( ) is the vector function b a F( = b a f ( ~ i + b a g( ~ j + b a h( ~ k t (ii) the indefinite integral of F ( ) is the vector function F() t = f() t i + g() t j + ht () k

18 Question 1 In questions 1(a) - 1(b), find F, F, F, F when t = 1. F () t = 2ti + t 2 + 3t j + 2t k (a) ( ) ( ) (b) ( t 1) ( 1 () 2 1 F t = e i + e j + t k ( ) 3 Question 2 d F G d F G In questions 2(a) - 2(b), find ( ) and ( ). (a) () t 2 Ft = ei+ j + tk, Gt () = t 3 i + j k F () t = t i t j + 2t+ 1 k, 2 ( ) (b) Gt () = 2t 3i + j tk ( )

19 Question 3 Find 2 2 F F F F F F,,,, 2 x y x y x x y xy ( ) ( ) F x, y = e i+ x y j+ xsin yk. for the given Question 4 Evaluate the integral in questions 4(a) and 4(b). (a) ( ) 2 t i t j+ k 0 (b) ( 3t ln ) 3 ti e j+ tk 1

20 4.3 Tangents Definition t F ( t 0 ) 0. Then ( ) Suppose F ( ) is differentiable at t0 and that F is defined to be a t 0 t tangent vector to the graph of F ( ) at the point where t = t 0. Unit tangent vector t If F ( ) is a vector function that defines a smooth graph, then at each point a unit tangent is Tt ( ) F () t = F () t and the principal unit normal vector is T () t Nt ( ) = T () t

21 Unit tangent vectors change direction along the curve, but always have length 1.

22 Smooth curve t The graph of the vector function defined by F ( ) is said to be smooth on any interval of t where F ( t ) is continuous and F ( 0. A curve that is smooth has a continuous turning tangent A curve that is not smooth can have sharp points. Note that this graph is piecewise smooth

23 Arc length Let rt () = fti () + gtj () + htk () be a differentiable vector valued function on [a, b]. Then the arc length s is defined by b s [ f ( ] [ g ( ] [ h ( ] = + + a b dx dy dz = + + a In a more compact form: ( ) ( ) ( ) b s = r () t a By FTC, ds = r () t

24 Example Find the length of the given curve:, 1 t e 2 r() t = t i + 2tj + lntk Solution The derivative has length Thus, 1 r () t = 2ti + 2j + t k 1 r () t = 2t + for t 1 t e 1 s t t ( ) arc length, = 2 + e 1 2 e 2 = t ln t + = ( e + 1) (1 + 0) 1 = e 2

25 Binormal Vector Binormal vector, T, N and B = T N B define a moving right handed vector frame, called Frenet frame or TNB frame play a role in calculating the paths of particles moving in space Question 1 In questions 1(a) - 1(b), find the unit tangent vector T, the principal unit normal vector, N the binormal unit at the indicated t. vector B, of the given rt () rt () = acosti + asin tj + btk, (a) ; ab > 0, t = π. (b) 2 rt = ti + t j+ tk t= (), 1.

26 Question 2 In questions 2(a) - 2(b), the coordinates of a moving particle are given as a function of time t. Find the speed v, the unit tangent vector T, as a function of t. (a) t t x = e cos t, y = e sin t, z = 0. (b) x = 5sin4 t, y = 5cos4 t, z = 10 t. Question 3 The position vector of a moving particle is t rt ( ) = ( sin t+ cos i+ ( sin t cos j+ k. 2 (a) Determine the velocity and speed of the particle. (b) Determine the acceleration of the particle. (c) Find a unit tangent to the path of the particle, in the direction of motion.