Vector Functions It is easy to see that (13.1.7) [f0+gt】di= f(tdr+g(t)dr and (13.1.8) [of(r)]dt =a f(t)dt for every constant scalar a. a It is also true that (13.1.9) [e.f(t)]dt =e f()dt for every constant vector c Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Vector Functions
(13.1.10) f()dt≤f(ll dt. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved
Differentiation Formulas What interests us here is that,if f,gand uare differentiable,then the newly constructed functions are also differentiable and their derivatives satisfy the following rules: (I)(f+gY(0=f'()+g() (2)(af)(t)=af'()(a constant). (3)(f)Y(0=ut0f'(t)+t(tf(0. (13.2.1) (4)(f.g'(t)=[f(t)·g(】+[f'(t)·g(t (5)(f×gY(t0)=[f(t)×g'(t】+[f'(t)×gt小. (6 (fou)'(1)=f'(u(t)u'()=u(1)f'(u())(the chain rule). Differentiation rules,p.772 Main Menu p5@
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Differentiation rules, p. 772
Example: f0=2r7-,80=i+万+rk0= Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Example: 2 2 3 3 1 f (t) = 2t i −3 j, g(t) = i + tj + t k , u(t) = t
Differentiation Formulas (1)(f+g)= df dg dt dt +dt d df (2②不af)= (a constant) (u =u(t)) dt (13.2.2) d 4) dg (df (f·g)=f. )+(·g df (5)(f×g)= f× dg dt dt 、g df df du (6) (chain rule) dt du dt Differentiation rules,Leibniz's notation,p.774 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Differentiation rules, Leibniz’s notation, p. 774
Differentiation Formulas dr dr (13.2.3) r dt (13.2.3),(13.2.4),Pp.774,775 Main Menu C00e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas (13.2.3), (13.2.4), pp. 774, 775
(13.2.4) 月=[×)×] Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved
Curves Suppose r(t)=x(t)i+y(t)j+z(t)k is differentiate on some interval I. Then r(t)defines a curve C We call C a differentiable curve and say that C is parametrized by r with parameter t. It is important to understand that the curve C is oriented in the sense that as t increase on I. Example:C:F(u)=cos(t)i+sin(t)j,tE[0,2] C2:F(u)=cos(2π-w)i+sin(2π-w)j,u∈[0,2π] Differentiable curves,etc.,p.776 Main Menu cme5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Curves Differentiable curves, etc., p. 776 Suppose is differentiate on some interval I. Then r(t) defines a curve C It is important to understand that the curve C is oriented in the sense that as t increase on I. Example: : ( ) cos(2 ) sin( 2 ) , [0,2 ] C2 r u = −u i + −u j u : ( ) cos( ) sin( ) , [0,2 ] C1 r u = t i + t j t
Curves DEFINITION 13.3.1 TANGENT VECTOR Let C:r(t)=x()i+y()j+z()k be a differentiable curve.The vector r'(t),if not 0,is tangent to the curve C at the point P(x(t).y(r),z()). + r(t+h h>0 Main Meny C00e
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Example:Find a point P on the curve 7(0）=(1-21)i+t2j+2e2-呢 at which the tangent vector r(t)is parallel to the radius vector r(t). Main Meny o8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Example: Find a point P on the curve at which the tangent vector is parallel to the radius vector . r t t i t j e k t 2 2( 1) ( ) (1 2 ) 2 − = − + + r(t) r'(t)