Prove that for orthogonal curvilinear coordinates, r e1 @ e2 @ e3 @ h1 @u1 h2 @u2 h3 @u3 in Visual Studio .NET

Drawing QR Code JIS X 0510 in Visual Studio .NET Prove that for orthogonal curvilinear coordinates, r e1 @ e2 @ e3 @ h1 @u1 h2 @u2 h3 @u3

Prove that for orthogonal curvilinear coordinates, r e1 @ e2 @ e3 @ h1 @u1 h2 @u2 h3 @u3
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[Hint: Let r a1 e1 a2 e2 a3 e3 and use the fact that d r dr must be the same in both rectangular and the curvilinear coordinates.] 7.89. Give a vector interpretation to the theorem in Problem 6.35 of 6.
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MISCELLANEOUS PROBLEMS 7.90. 7.91. 7.92. If A is a di erentiable function of u and jA u j 1, prove that dA=du is perpendicular to A. Prove formulas 6, 7, and 8 on Page 159. If  and  are polar coordinates and A; B; n are any constants, prove that U n A cos n B sin n satis es Laplace s equation. If V 2 cos  3 sin3  cos  , nd r2 V. r2 Ans. 6 sin  cos  4 5 sin2  r4
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7.93. 7.94.
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Find the most general function of (a) the cylindrical coordinate , (b) the spherical coordinate r, (c) the spherical coordinate  which satis es Laplace s equation. Ans. (a) A B ln ; b A B=r; c A B ln csc  cot  where A and B are any constants. Let T and N denote respectively the unit tangent vector and unit principal normal vector to a space curve r r u , where r u is assumed di erentiable. De ne a vector B T N called the unit binormal vector to the space curve. Prove that
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VECTORS
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These are called the Frenet-Serret formulas and are of fundamental importance in di erential geometry. In these formulas  is called the curvature,  is called the torsion; and the reciprocals of these,  1= and  1=, are called the radius of curvature and radius of torsion, respectively. 7.96. (a) Prove that the radius of curvature at any point of the plane curve y f x ; z 0 where f x is di erentiable, is given by    1 y02 3=2         y 00 (b) Find the radius of curvature at the point =2; 1; 0 of the curve y sin x; z 0. p Ans. (b) 2 2 7.97. Prove that the acceleration of a particle along a space curve is given respectively in (b) spherical coordinates by _ __  2 e  2 e zez _ __ _ _ r2 r2 sin2  er r 2_ r2 sin  cos  e 2_ sin  2r cos  r sin  e r r_ r_ where dots denote time derivatives and e ; e ; ez ; er ; e ; e are unit vectors in the directions of increasing ; ; z; r; ; , respectively. 7.98. 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 r E 0; r H 0; r E 1 @H ; c @t r H 1 @E c @t 1 (a) cylindrical,
prove that E and H satisfy the equation r2 1 @2 c2 @t2 2
where is a generic meaning, and in particular can represent any component of E or H. [The vectors E and H are called electric and magnetic eld vectors in electromagnetic theory. Equations (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.] 7.99. Use the relations in Problem 7.98 to show that @ 1 2 f E H 2 g cr E H 0 @t 2 7.100. Let A1 ; A2 ; A3 be the components of vector A in an xyz rectangular coordinate system with unit vectors 0 0 0 i1 ; i2 ; i3 (the usual i; j; k vectors), and A1 ; A2 ; A3 the components of A in an x 0 y 0 z 0 rectangular coordinate system which has the same origin as the xyz system but is rotated with respect to it and has the unit vectors 0 0 0 i1 ; i2 ; i3 . Prove that the following relations (often called invariance relations) must hold:
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