# (a) By Problem 7.43, r r f r f 0 r r r in .NET framework Creating QR Code 2d barcode in .NET framework (a) By Problem 7.43, r r f r f 0 r r r

(a) By Problem 7.43, r r f r f 0 r r r
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By Problem 7.35, assuming that f r has continuous second partial derivatives, we have & 0 ' f r r Laplacian of  r2  r r r r & 0 ' & ' f r f 0 r 1 d f 0 r f 0 r r r r 3 r r r r r r dr r r 00 0 0 r f r f r 2 3 f r 2 r f 00 r f 0 r r r r3 Another method: In spherical coordinates, we have     1 @ 2 @U 1 @ @U 1 @2 U r2 U 2 r 2 sin  2 2 @r @ r @r r sin  @ r sin  @2
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If U f r , the last two terms on the right are zero and we nd r2 f r 1 d 2 0 2 r f r f 00 r f 0 r r r2 dr
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VECTORS
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(b) From the result in part (a), we have r2       1 d2 1 2 d 1 2 2 2 3 3 0 r r dr r dr r r r
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showing that 1=r is a solution of Laplace s equation.
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7.45. A particle moves along a space curve r r t , where t is the time measured from some initial time. If v jdr=dtj ds=dt is the magnitude of the velocity of the particle (s is the arc length along the space curve measured from the initial position), prove that the acceleration a of the particle is given by a dv v2 T N dt 
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where T and N are unit tangent and normal vectors to the space curve and   1 8 ! !2 !2 9 1=2 < d 2x 2 d 2 r d2y d 2z =     2  ds  : ds2 ds2 ds2 ;
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The velocity of the particle is given by v vT. a Then the acceleration is given by 1
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dv d dv dT dv dT ds dv dT vT T v T v T v2 dt dt dt dt dt ds dt dt ds Then di erentiating with respect to s, or T dT 0 ds
Since T has a unit magnitude, we have T T 1. T dT dT T 0; ds ds 2T
dT 0 ds
from which it follows that dT=ds is perpendicular to T. Denoting by N the unit vector in the direction of dT=ds, and called the principal normal to the space curve, we have dT N ds where  is the magnitude of dT=ds. dT=ds d 2 r=ds2 . Hence 2
Now since T dr=ds [see equation (7), Page 157], we have
  8 ! !2 !2 91=2 d 2 r  < d 2 x 2 d2y d2z =     2  ds  : ds2 ds2 ds2 ; De ning  1=, (2) becomes dT=ds N=. a Thus from (1) we have, as required,
dv v2 T N  dt
The components dv=dt and v2 = in the direction of T and N are called the tangential and normal components of the acceleration, the latter being sometimes called the centripetal acceleration. The quantities  and  are respectively the radius of curvature and curvature of the space curve.
VECTORS
[CHAP. 7
Supplementary Problems
VECTOR ALGEBRA 7.46. 7.47. Given any two vectors A and B, illustrate geometrically the equality 4A 3 B A A 3B. A man travels 25 miles northeast, 15 miles due east, and 10 miles due south. By using an appropriate scale, determine graphically (a) how far and (b) in what direction he is from his starting position. Is it possible to determine the answer analytically Ans. 33.6 miles, 13.28 north of east. If A and B are any two non-zero vectors which do not have the same direction, prove that mA nB is a vector lying in the plane determined by A and B. If A, B, and C are non-coplanar vectors (vectors which do not all lie in the same plane) and x1 A y1 B z1 C x2 A y2 B z2 C, prove that necessarily x1 x2 ; y1 y2 ; z1 z2 . Let ABCD be any quadrilateral and points P; Q; R; and S the midpoints of successive sides. Prove (a) that PQRS is a parallelogram and (b) that the perimeter of PQRS is equal to the sum of the lengths of the diagonals of ABCD. Prove that the medians of a triangle intersect at a point which is a trisection point of each median. Find a unit vector in the direction of the resultant of vectors A 2i j k, B i j 2k, C 3i 2j 4k. p Ans. 6i 2j 7k = 89
7.51. 7.52.
THE DOT OR SCALAR PRODUCT 7.53. 7.54. Evaluate j A B A B j if A 2i 3j 5k and B 3i j 2k. Ans. 24
Verify the consistency of the law of cosines for a triangle. [Hint: Take the sides of A; B; C where C A B. Then use C C A B A B .] Find a so that 2i 3j 5k and 3i aj 2k are perpendicular. Ans. a 4=3
7.55. 7.56.
If A 2i j k; B i 2j 2k and C 3i 4j 2k, nd the projection of A C in the direction of B. Ans. 17/3 A triangle has vertices at A 2; 3; 1 ; B 1; 1; 2 ; C 1; 2; 3 . Find (a) the length of the median drawn from B to side AC p and (b) the acute angle which this median makes with side BC. p Ans. (a) 1 26; b cos 1 91=14 2 Prove that the diagonals of a rhombus are perpendicular to each other. Prove that the vector AB BA = A B represents the bisector of the angle between A and B.
7.58. 7.59.
THE CROSS OR VECTOR PRODUCT 7.60. 7.61. If A 2i j k and B i 2j 3k, nd j 2A B A 2B j: Ans.
p 5 3
Find a unit vector p perpendicular to the plane of the vectors A 3i 2j 4k and B i j 2k. Ans. 2j k = 5 If A B A C, does B C necessarily Find the area of the triangle with vertices 2; 3; 1 ; 1; 1; 2 ; 1; 2; 3 . Ans.
7.62. 7.63.
p 3
CHAP. 7]