Page 189 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 189
180 VECTORS [CHAP. 7
JACOBIANS AND CURVINLINEAR COORDINATES
@r @r
@ðx; y; zÞ @r
7.84. Prove that .
¼
@u 1 @u 2 @u 3
@ðu 1 ; u 2 ; u 3 Þ
2
7.85. Express (a)grad ; ðbÞ div A; ðcÞr in spherical coordinates.
@ 1 @ 1 @
Ans: e 3
@r e 1 þ r @ e 2 þ r sin @
ðaÞ
1 @ 2 1 @ 1 @A 3
where A ¼ A 1 e 1 þ A 2 e 2 þ A 3 e 3
r @r r sin @ r sin @
ðbÞ 2 ðr A 1 Þþ ðsin A 2 Þþ
2
1 @ 2 @ 1 @ 1 @
r sin
r @r @r r sin @ @ r sin @
ðcÞ 2 þ 2 þ 2 2 2
7.86. The transformation from rectangular to parabolic cylindrical coordinates is defined by the equations
2
2
2
1
x ¼ ðu v Þ, y ¼ uv, z ¼ z. (a)Prove that the system is orthogonal. (b)Find ds and the scale
2
factors. (c)Find the Jacobian of the transformation and the volume element.
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
2
2
2
2
2
2
ds ¼ðu þ v Þ du þðu þ v Þ dv þ dz ; u þ v ; h 3 ¼ 1
Ans. ðbÞ h 1 ¼ h 2 ¼
2
2
2
2
u þ v ; ðu þ v Þ du dv dz
ðcÞ
2
7.87. Write (a) r and (b)div A in parabolic cylindrical coordinates.
!
2
2
2
1 @ @ @
2
Ans: ðaÞ r ¼ þ þ
2
u þ v 2 @u 2 @v 2 @z 2
1 @ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @A 3
2 2 2 2
u þ v @u @v @z
ðbÞ div A ¼ 2 2 ð u þ v A 1 Þþ ð u þ v A 2 Þ þ
7.88. Prove that for orthogonal curvilinear coordinates,
e 1 @ e 2 @ e 3 @
r ¼ þ þ
h 1 @u 1 h 2 @u 2 h 3 @u 3
[Hint: Let r ¼ a 1 e 1 þ a 2 e 2 þ a 3 e 3 and use the fact that d ¼r 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 Chapter 6.
MISCELLANEOUS PROBLEMS
7.90. If A is a differentiable function of u and jAðuÞj ¼ 1, prove that dA=du is perpendicular to A.
7.91. Prove formulas 6, 7, and 8 on Page 159.
n
7.92. If and are polar coordinates and A; B; n are any constants, prove that U ¼ ðA cos n þ B sin n Þ
satisfies Laplace’s equation.
3
2cos þ 3 sin cos 2 2
7.93. If V ¼ , find r V. Ans. 6 sin cos ð4 5 sin Þ
r 2 r 4
7.94. Find the most general function of (a)the cylindrical coordinate ,(b)the spherical coordinate r,(c)the
spherical coordinate which satisfies Laplace’s equation.
Ans.(a) A þ B ln ; ðbÞ A þ B=r; ðcÞ A þ B lnðcsc cot Þ where A and B are any constants.
7.95. Let T and N denote respectively the unit tangent vector and unit principal normal vector to a space curve
r ¼ rðuÞ, where rðuÞ is assumed differentiable. Define a vector B ¼ T N called the unit binormal vector to
the space curve. Prove that