CHAPT E R 45
Vector Functions in Space . Divergence and Curl . Line Integrals 45.1
For the space curve R(t) = (t, t 2, t3 ) , find a tangent vector and the tangent line at the point ( 1 , 1 , 1 ) .
I A tangent vector i s given b y the derivative R'(t) = ( 1 , 2t, 3t 2 ) . A t ( 1 , 1 , 1 ) , t = l . Hence , a tangent vector is ( 1 , 2, 3). Parametric equations for the tangent line are x 1 + u, y = 1 + 2u , z = 1 + 3u . As a vector function , the tangent line can be represented by ( 1 , 1 , 1 ) + u( l , 2 , 3 ) .
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45.2
Find the speed o f a particle tracing the curve o f Problem 4 5 . 1 a t time t = 1 . (The parameter t is usually , but not necessarily, interpreted as the time.)
� = �( �r + ( drr + ( �;r is the speed. In general, if R(t) = ( dx dy dz ) ds , , and , , (x(t) , y(t), z(t)) , R (t) = I R (t) I . For this particular case , � I ( 1 , 2t, 31 2 ) ! = dt dt dt V l 4t 2 9t4• When t �� I If s is the arc length , ,
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45.3
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Find the normal plane to the curve of Problem 45 . 1 at t
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I The tangent vector at t = 1 is ( 1 , 2, 3 ) . That vector is perpendicular to the normal plane . Since the normal plane contains the point ( 1 , 1 , 1 ) , its equation is l (x - 1) + 2( y - 1 ) + 3(z - 1 ) = 0, or equivalently, x + 2y + 3 z = 6. 45.4
Find a tangent vector, the tangent line, the speed , and the normal plane to the helical curve R(t) = (a cos 2 7rt, a sin 2 1Tt, bt) at t l . =
I A tangent vector is R'(t) = ( - 2 1Ta sin 21Tt, 21Ta cos 21Tt, b ) (0, 21Ta, b ) . The tangent line is (a, 0 , b ) + u(O, 21Ta, b ) . The speed is IR'(t)I = v'4 7r 2a 2 + b 2• The normal plane has the equation (O)(x - a) + (2 7ra)( y - 0) + (b)(z - b ) 0, or equivalently , 21Tay + bz b 2• =
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45.5
Prove that the angle 8 between a tangent vector and the positive z-axis is the same for all points of the helix of Problem 45.4.
I R '(t) = (-21Ta sin 21Tt, 2 1Ta cos 2 1Tt, b) has a constant z-component ; thus, 8 is constant. 45.6
Find a tangent vector, the tangent line , the speed, and the normal plane to the curve R(t) = (t cos t, t sin t, t) at t = 7T/2. I R'(t) = (cos t - t sin t, sin t + t cos t, 1 ) = (- 7T/2, l , 1 ) is a tangent vector when t = 1Tl2. The tangent line is traced out by (0, 7T/2, 7T/2) + u(- 1Tl2, 1 , 1 ) , that is, x = (- 7T/2)u, y = 7T / 2 + u , z = 7T/2 + u . The speed at t = 7T / 2 is ds /dt = IR'(t)I = v' 7T 2/4 + 2 !V 7T 2 8. An equation of the normal plane is (- 7T/2)(x - 0) + ( y - 7T/2) + (z - 7T/2) = 0, or equivalently, (- 7T/2)x + y + z = 1T. =
45. 7
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If G(t) = (t, t 3, In t) and F(t) = t 2 G(t) , find F'(t) .
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In general, [ g(t)G(t)] = g(t)G'(t) + g'(t)G(t) . (The proof in Problem 34.53 is valid for arbitrary vector functions.) Henc� . F'(t) = t 2 ( 1 , 3t 2, l it) + 2t(t, t3, In t) = (3t 2, 5t 4, t + 2t In t).
45.8
If G(t) = ( e', cos t, t) , find
*' [(sin t)G(t)] . By the formula of Problem 45 .7, *' [(sin t)G(t)] = (sin t)G'(t) + (: sin t) G(t) = (sin t)(e', - sin t , 1) + t 2 2
I (cos t)(e', cos t, t) = (e'(sin t + cos t), cos t - sin t, sin t + t cos t) .
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