LineIntegral_VectorField

LineIntegral_VectorField - 824 CHAPTER 15 Vector Fields...

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824 CHAPTER 15 Vector Fields Exercises 15.3 1. Show that the curve C given by r = a cos t sin t i + a sin 2 t j + a cos t k ,( 0 t π 2 ), lies on a sphere centred at the origin. Find Z C zds . 2. Let C be the conical helix with parametric equations x = t cos t , y = t sin t , z = t , ( 0 t 2 π) .F ind Z C . 3. Find the mass of a wire along the curve r = 3 t i + 3 t 2 j + 2 t 3 k 0 t 1 ), if the density at r ( t ) is 1 + t g/unit length. 4. Show that the curve C in Example 3 also has parametrization x = cos t , y = sin t , z = cos 2 t , ( 0 t π/ 2 ) ,and recalculate the mass of the wire in that example using this parametrization. 5. Find the moment of inertia about the z -axis (i.e., the value of δ Z C ( x 2 + y 2 ) ds ), for a wire of constant density δ lying along the curve C : r = e t cos t i + e t sin t j + t k , from t = 0 to t = 2 π . 6. Evaluate Z C e z ,where C is the curve in Exercise 5. 7. Find Z C x 2 along the line of intersection of the two planes x y + z = 0and x + y + 2 z = 0, from the origin to the point ( 3 , 1 , 2 ) . 8. Find Z C p 1 + 4 x 2 z 2 C is the curve of intersection of the surfaces x 2 + z 2 = 1and y = x 2 . 9. Find the mass and centre of mass of a wire bent in the shape of the circular helix x = cos t , y = sin t , z = t , ( 0 t 2 , if the wire has line density given by δ( x , y , z ) = z . 10. Repeat Exercise 9 for the part of the wire corresponding to 0 t π . 11. Find the moment of inertia about the y -axis, that is, Z C ( x 2 + z 2 ) , of the curve x = e t , y = 2 t , z = e t , ( 0 t 1 ) . 12. Find the centroid of the curve in Exercise 11. 13. 3 Find Z C xds along the ±rst octant part of the curve of intersection of the cylinder x 2 + y 2 = a 2 and the plane z = x . 14. 3 Find Z C along the part of the curve x 2 + y 2 + z 2 = 1, x + y = 1, where z 0. 15. 3 Find Z C ( 2 y 2 + 1 ) 3 / 2 C is the parabola z 2 = x 2 + y 2 , x + z = 1. Hint: Use y = t as parameter. 16. Express as a de±nite integral, but do not try to evaluate, the value of Z C xyzds C is the curve y = x 2 , z = y 2 from ( 0 , 0 , 0 ) to ( 2 , 4 , 16 ) . 17. 3 The function E ( k ,φ) = Z φ 0 p 1 k 2 sin 2 tdt is called the elliptic integral function of the second kind .The complete elliptic integral of the second kind is the function E ( k ) = E ( k ,π/ 2 ) . In terms of these functions, express the length of one complete revolution of the elliptic helix x = a cos t , y = b sin t , z = ct , where 0 < a < b . What is the length of that part of the helix lying between t = t = T ,where0 < T <π/ 2? 18. 3 Evaluate Z L x 2 + y 2 L is the entire straight line with equation Ax + By = C , ( C 6= 0 ) . Hint: Use the symmetry of the integrand to replace the line with a line having a simpler equation but giving the same value to the integral. 15.4 Line Integrals of Vector Fields In elementary physics the work done by a constant force of magnitude F in moving an object a distance d is de±ned to be the product of F and d : W = Fd .T h e r e is, however, a catch to this; it is understood that the force is exerted in the direction of motion of the object. If the object moves in a direction different from that of the force (because of some other forces acting on it), then the work done by the particular
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This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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LineIntegral_VectorField - 824 CHAPTER 15 Vector Fields...

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