Exercises

Exercises
1. Find 2222
,,,d d d d dt dt dt dt r r r r , if ()sin cos t t t t =++r i j k . 2. Show that d d A dt dt
⋅=⋅A A A . 3. A particle moves along the curve 22()2(4)(35)t t t t t =+−+−r i j k , where is time. Find
its velocity and acceleration at .
t 1t =4. A velocity field is given by 2
24x xy t =−+v i j k . Determine the acceleration at the point .
(2,1,4)−5. Find the arc length of ()cos sin t a t a t bt =++r i j k from 0t = to 2t π=.
6. Find the gradient of the scalar field xyz φ=, and evaluate it at the point , find the derivative of (1,2,3)φ in the direction of +i j .
7. The temperature is given by T 2T x xy yz =++. What is the unit vector that points in the
direction of maximum change of temperature at ? What is the value of the derivative of the temperature in (2,1,4)x direction at that point?
8. Find the unit normal to each of the following surfaces at the point indicated, (a) at , (b) 220x y z +−=(1,1,2)225x y +=at , and (c) (2,1,0)23y x z =+at .
(1,2,1)9. Find the divergence of each of the following vector fields at the point .
(2,1,1)−(a) 22x yz y =++F i j k ,
(b) x y y =++F i j k , (c) (
r x y y ==++F r i j k
10. Verify the divergence theorem by calculating both the volume integral and the surface integral for the vector field ()y x z x =++−F i j k and the volume of the unit cube 0,,1x y z ≤≤.
11. By using the divergence theorem, evaluate
(a)
()S x y z da ++⋅∫∫i j k n w , (b) ()2
S
x x z da ++⋅∫∫i j k n w ,
where S is the surface of the cylinder 224,08;x y z +=≤≤
(c) ()2sin cos sin ,S x y x z y ++⋅∫∫i da j k n w
where S is the surface of the sphere 222(2)x y z 1++−=.
12. Show that
3S
da V ⋅=∫∫r n w , where V is the volume bounded by the closed surface S.
13. Recognizing that ; da dydz ⋅=i n da dxdz ⋅=j n ; da dydy ⋅=k n , evaluating the following integral using the divergence theorem
()S
xdydz ydxdz zdxdy ++∫∫w , where S is the surface of the cylinder 22
9,0 3.x y z +=≤≤
14. Evaluating the following integral using the divergence theorem ()2
2S
xdydz ydxdz y dxdy ++∫∫w , where S is the surface of the sphere 2224x y z ++=.
15. Find the curl of each of the following vector fields at the point (2,4,1)−.
(a) 222x y z =++F i j k and
(b) 2xy y xz =++F i j k 16. Verify Stokes’ theorem by evaluating both the line and surface integral for the vector field 22(2)x y y y z =−−+A i j k and the surface S given by the disc 220,1z x y =+≤.
17. Show that 0C d ⋅=∫r r v for any closed curve C.
18. Calculate the circulation of the vector 22y xy z =++F i j k ()d ⋅∫F r v around a triangle with
vertices at the origin, (2, 2, 0), and (0, 2, 0) by (a) direct integration, and (b) using Stokes’ theorem.
19. Calculate the circulation of y x z =−+F i j k around a unit circle in the xy plane with center at the origin by (a) direct integration and (b) using Stokes’ theorem.
20. Check the product rule
()()()∇⋅×=∇×⋅−∇×⋅A B A B B A
by calculating each term separately for the functions 222,y xy z =++A i j k 3sin sin y x z =++B i j k .
21. Check the relation
()()2∇×∇×=∇∇⋅−∇A A A
by calculating each term separately for the function 22
2y xy z =++A i j k .
(hint: 2x y x A A A ∇=Δ=Δ+Δ+ΔA A i j k )
22. Show that
()0φφ∇×∇=23. Show that for any closed surface S ()0S
da ∇×⋅=∫∫B n w 24. For what values, if any, of the constants a and b is the following vector field irrotational?
2(cos )(sin )()y x axz b x z x y =+++++F i j k
25. (a) Show that 2(23)(4)4xy x z =++−−F i y j k is an irrotationa field.
(b) Find a scalar potential
φ such that φ∇=−F . (c) Evaluate the integral
2,1,13,1,2d −−⋅∫F r . 26. Show that if in the volume V , then
20φ∇=0S
da φ∇⋅=∫∫n w where S is the surface bounding the volume.。