Problem Set 4   -   Solutions

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1. Convert   F   from Cartesian coordinates to spherical polar coordinates.

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   Let   c₁ = cos q   ,       s₁ = sin q
           c₂ = cos f   and   s₂ = sin f ,   then
   x   =   r s₁c₂     and     y   =   r s₁s₂
   Þ     x² + y²   =   r²s₁² (c₂² + s₂²)   =   r²s₁²
   Þ       =   r s₁
   [Note:     for all values in the spherical polar (r, q, f) system.
   There is therefore no ambiguity in the square root.]
   
   
   

   

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2. Relative to the coordinate axes, in what direction is   F   pointing?

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   is parallel to the x-y plane everywhere in
   (except on the z-axis, where F is undefined)

   In the horizontal plane   z = r cos q :
   F_y = sin f
   F_x = cos f       (from part (a) above)
   
   Þ    

   OR

   In the vertical plane containing the point P (r, q, f) and the z-axis
   (which is the plane   f = constant):

   The vertical component of F is
   F_r cos q   -   F_q sin q
   =   sin q cos q   -   cos q sin q   =   0
   Therefore F is parallel to the x-y plane.

   The azimuthal component (perpendicular into the paper) is
   -F_x sin f   +   F_y cos f
   =   - cos f sin f   +   sin f cos f   =   0

   F is therefore in the same vertical plane as the point P and the z-axis.

   Þ    

The coordinate conversion matrix   A   for Cartesian coordinates to spherical polar coordinates is

so that   F_sph   =   A F_cart   where

Show that the conversion matrix   B   for the inverse conversion from spherical polar back to Cartesian coordinates, such that   F_cart = B F_sph,   is simply

B = A^T       (the transpose of matrix   A).

B = A^-1   (because the two transformations are inverses of each other)
In order to show that A^-1 = A^T, it is sufficient to show that A^TA = I.
[This is much faster than the adjoint/determinant method or the row reduction method for finding A^-1.]

  [and the coordinate transformation matrix A is orthogonal.]

Convert from spherical polar to Cartesian coordinates the vector field

From Question 2, B = A^T.
Use the same abbreviations as in questions 1 and 2 :
Let   c₁ = cos q   ,       s₁ = sin q
        c₂ = cos f   and   s₂ = sin f ,   then
F_r   =   c₁   ,     F_q   =   -s₁   ,       F_f   =   0

Find the divergence and curl of each of the following:

1. 

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                 =   y² + z² + x²         Therefore

   

   

   

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2. 

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   [Note:   the cyclic symmetry between x, y, z   in parts (a) and (b) above allows any two parts of the calculations for divergence and curl to be deduced from the other part.]

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3. 

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For the vector field defined in spherical polar coordinates by

find   .

Recall, from page 1-38 of the lecture notes, that
       
       
and  
Use the abbreviations


Using the product and chain rules of differentiation,


Therefore, everywhere in space and at all values of t,

OR

Convert the vector u from spherical polar coordinates to Cartesian coordinates:

Consider the purely radial vector field   F (r, q, f) = f (r) , where is the unit radial vector in the spherical polar coordinate system and   f (r)   is any function of r that is differentiable everywhere in (except possibly at the origin).

1. Find an expression, in terms of   r,   f (r)   and   f ' (r), for the divergence of F.

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   First note that   f (r)   is a function of   r   only, so that
   
   
   Therefore, everywhere except possibly on the z axis,

   

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2. Find an expression, in terms of   r,   f (r)   and   f ' (r), for the curl of F.

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   [All purely radial differentiable vector fields are therefore automatically irrotational.]

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3. Of particular interest is the central force law

   

   Show that the divergence of   F   vanishes everywhere in (except possibly at the origin) if and only if   n = 2 .
   [Two of the four fundamental forces of nature, electromagnetism and gravity, both obey this inverse square law.]

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   div F   therefore vanishes for all   r > 0   if and only if   n = 2 .

   [Note:   n = 2   recovers the familiar inverse square law,
                   ]