CURVILINEAR COORDINATES AND JACOBIANS
5.42. Express in cylindrical coordinates: (a) grad *, (&) divA, (c)
Let M! = p, M2 = 0, w3 = z, fi! = 1, fe2 = p, A3 = 1 [see Problem 5.40(a)] in the results 1, 2, page 128, and 4, page 129. Then
(a) grad * =
where e1( e2, e3 are the unit vectors in the directions of increasing p, <f>, z respectively.
(6) divA = =
where A = A1e1 + A2e2 + A3es.
(e)
5.43. If F(x,y,u,v) = Q and G(x, y, u, v) = 0, find (a) du/dx, (b) du/dy, (c) dv/dx, (d) dv/dy.
The two equations in general define the dependent variables u and v as (implicit) functions of the independent variables x and y. Using the subscript notation, we have
(1) dF = Fxdx + Fvdy + Fudu + Fvdv = 0 (2) dG = Gxdx + Gydy + Gndu + Gvdv = 0 Also, since u and v are functions of x and y,
(S) du — uxdx + uvdy (4) dv = vxdx + vydy Substituting (S) and (4) in (1) and (2) yields
(5) dF = (Fx + Fuux + Fvvx) dx + (Fy + Fuuy + f>9) dy = 0 (6) dG = (Gx + Guux + Gvvx) dx + (Gy + Guuy + Gvvy) dy = 0
Since * and y are independent, the coefficients of dx and dy in (5) and (6) are zero. Hence we obtain
M !>.«» + F,v, = -Fx (Fuuy + Fvvy = -Fv V> \Guux + Gcvx = -Gx * ; \Guuy + Gvvy = -Gy
Solvinsr (7\ and (8\ arives
Supplementary Problems
VECTOR ALGEBRA
5.44. Given any two vectors A and B, illustrate geometrically the equality 4A + 3(B — A) = A + 3B.
5.45. 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 (6) in what direction he is from his starting position.
Is it possible to determine the answer analytically?
5.46. If A and B are any two non-zero vectors which do not have the same direction, prove that mA + wB is a vector lying in the plane determined by A and B.
Fu Fv The functional determinant ,. _ , denoted b y . i s the Jacobian of G
u Gv
F and G with respect to u and v and is supposed ¥* 0.
Note that it is possible to devise mnemonic rules for writing at once the required partial deriva- tives in terms of Jacobians.
denoted by i s t h e j a s o b i o n
5.47. If A, B and C are non-coplanar vectors (vectors which do not all lie in the same plane) and a^A + 2/jB + «tC = x2A + j/2B + Z2C, prove that necessarily Xi = x2, yt = yz, «i = ^2-
5.48. 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 (6) that the perimeter of PQRS is equal to the sum of the lengths of the diagonals of ABCD.
5.49. Prove that the medians of a triangle intersect at a point which is a trisection point of each median.
5.50. Find a unit vector in the direction of the resultant of vectors A = 2i — j + k, B = i + j + 2k, C = 3i - 2j + 4k.
THE DOT OR SCALAR PRODUCT
5.51. Evaluate |(A + B) • (A - B)| if A = 2i - 3j + 5k and B = 3i + j - 2k.
5.52. Prove the law of cosines for a triangle. [Hint. Take the sides as A, B, C where C = A — B. Then use C - C = (A-B)-(A-B).]
5.53. Find a so that 2i — 3j + 5k and 3i + oj — 2k are perpendicular.
5.54. If A = 2i + j + k, B = i - 2j + 2k and C = 3i - 4j + 2k, find the projection of A + C in the direction of B.
5.55. A triangle has vertices at A(Z, 3,1), B(—l, 1,2), C(l, —2,3). Find (a) the length of the median drawn from B to side AC and (6) the acute angle which this median makes with side BC.
5.56. Prove that the diagonals of a rhombus are perpendicular to each other.
5.57. Prove that the vector (AB + BA)/(A + B) represents the bisector of the angle between A and B.
THE CROSS OR VECTOR PRODUCT
5.58. If A = 2i-j + k and B - i + 2j-3k, find |(2A + B) X (A- 2B)|.
5.59. Find a unit vector perpendicular to the plane of the vectors A = 3i — 2j 4- 4k and B = i + j — 2k.
5.60. If A X B = A X C, does B = C necessarily?
5.61. Find the area of the triangle with vertices (2, -3,1), (1, -1,2), (-1, 2, 3).
5.62. Find the shortest distance from the point (3,2,1) to the plane determined by (1,1,0), (3, —1,1), (-1,0,2).
TRIPLE PRODUCTS
5.63. If A = 2i + j-3k, B = i-2j + k, C = -i + j-4k, find (a) A • (B X C), (6) C • (A X B), (c) A X (B X C), (d) (A X B) X C.
5.64. Prove that (a) A • (B X C) = B • (C X A) = C • {A X B) (6) A X (B X C) = B(A • C) - C(A • B).
5.65. Find an equation for the plane passing through (2, —1, —2), (—1,2, —3), (4,1,0).
5.66. Find the volume of the tetrahedron with vertices at (2,1,1), (1, -1, 2), (0,1, -1), (1, -2,1).
5.67. Prove that (A X B) • (C X D) + (B X C) • (A X D) + (C X A) • (B X D) = 0.
DERIVATIVES
5.68. A particle moves along the space curve r = e~* cos t i + e~* sin t j + e-*k. Find the magnitude of the (a) velocity and (6) acceleration at any time t.
5.69. Prove that where A and B are differentiable functions of u.
5.70. Find a unit vector tangent to the space curve * = t, y = t2, z = i3 at the point where í = 1.
5.71. If r = a cos at + b sin at, where a and b are any constant noncollinear vectors and « is a constant scalar, prove that (a) r X
5.72. If A = X2i -yj + xzk, B = yi + xj- xyzk and C = i - yj + xszk, find ( a ) a n d (6) d[A • (B X C)] at the point (1, -1,2).
5.74. If A is a differentiate function of u and |A(tt)| = 1, prove that dA/du is perpendicular to A.
5.75. Let T and N denote respectively the unit tangent vector and unit principal normal vector to a space curve r = i(u), where r(u) is assumed differentiate. Define a vector B = T X N called the unit binormal vector to the space curve. Prove that
These are called the Frenet-Serret formulas. In these formulas K is called the curvature, T is called the torsion; and the reciprocals of these, p — l//c and a = I/T, are called the radius of curvature and radius of torsion respectively.
GRADIENT, DIVERGENCE AND CURL
5.76. If U, V, A, B have continuous partial derivatives prove that:
(a)
5.77. If <t> = xy + yz + zx and A = x*yi + y2zj + z2xk, find at the point (3, —1,2).
5.78. Show that = 0 where r = xi + yj + zk and r = |r|.
5.79. Prove:
5.80. Prove that curl grad U = 0, stating appropriate conditions on U.
5.81. Find a unit normal to the surface xzy — 2xz + 2?/2z4 = 10 at the point (2,1, —1).
5.82. If A = 3xz*i - yzj + (x + 2z)k, find curl curl A.
5.83. (a) Prove that ' (6) Verify the result in (a) if A is given as in Problem 5.82.
5.84. Find the equations of the (a) tangent plane and (b) normal line to the surface x2 + y2 = 4z at (2,-4, 5).
5.85. Find the equations of the (a) tangent line and (6) normal plane to the space curve x = 6 sin t, y = 4 cos 3t, z = 2 sin 5t at the point where t = ir/4.
5.86. (a) Find the directional derivative of U = 2xy — z2 at (2, —1,1) in a direction toward (3,1, —1).
(6) In what direction is the directional derivative a maximum? (c) What is the value of this maximum?
5.87. Prove that the acute angle y between the z axis and the normal to the surface F(x, y, z) = 0 at any point is given by sec y =
5.88. (a) Develop a formula for the shortest distance from a point (x0, y0, z0) to a surface. (6) Illustrate the result in (a) by finding the shortest distance from the point (1,1, —2) to the surface z = xz + yz.
5.73. If R at the point
5.89. Let E and H be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations
U) prove that E and H satisfy the equation
(2) [The vectors E and H are called electric and magnetic field vectors in electromagnetic theory. Equa- tions (1) are a special case of Maxwell's equations. The result (2) led Maxwell to the conclusion that light was an electromagnetic phenomena. The constant c is the velocity of light.]
5.90. Use the relations in Problem 5.89 to show that
JACOBIANS AND CURVILINEAR COORDINATES 5.91. Prove that
5.92. Express (a) grad *, (6) div A in spherical coordinates.
5.93. The transformation from rectangular to parabolic cylindrical coordinates is defined by the equations x = £(w.2 — vz), y = uv, z = z. (a) Prove that the system is orthogonal. (6) Find ds2 and the scale factors, (c) Find the Jacobian of the transformation and the volume element.
5.94. Write (a) V2* and (6) div A in parabolic cylindrical coordinates.
5.95. Prove that for orthogonal curvilinear coordinates, V* =
[Hint. Let V* = ct1e1 + <t2e2 + a3es and use the fact that d<I> = V* • dr must be the same in both rectangular and the curvilinear coordinates.]
5.96. Prove that the acceleration of a particle along a space curve is given respectively in (a) cylindrical, (6) spherical coordinates by
(p- P^)ep + (p* + 2¿¿fe» + '¿9.
( V — rez — rj>2 sin2 e)er + (r 'ê + 2re — rj>2 sin e cos 0)ee + (2f 0 sin e + 2rèj> cos e + r 0 sin e)e^, where dots denote time derivatives and ep, e^, ez, er, ee, e,,, are unit vectors in the directions of increasing p, 0, z, r, 9, 0 respectively.
5.97. If F = x + 3y2 - zs, G - 2x2yz, and H = 2z2 — xy, evaluate at (1, -1,0).
5.98. If F = xy + yz + zx, G = a;2 + j/2 + z2, and H = x + y + z, determine whether there is a functional relationship connecting F, G, and H, and if so find it.
5.99. If F(P, V, T) = 0, prove that (a) = -1 where a subscript indicates the variable which is to be held constant. These results are useful in thermo- dynamics where P, V, T correspond to pressure, volume and temperature of a physical system.
d(u'v'w) * °' ^ Give an interPretation of the result of (a) in terms of transformations.
5.10.0
Answers to Supplementary Problems
5.45. 33.6 miles, 13.2° north of east 5.50.
5.51.
5.53.
5.54.
5.55.
5.58.
5.59.
5.61.
5,62.
5.63.
5.65.
5.66.
5.68.
5.70.
5.72.
5.73.
5.77.
5.81.
5.82.
5.84.
5.85.
5.86.
5.92.
5.93.
5.94.
5.97.
5.98.