When to use the Jacobian in spherical coordinates?

  • Thread starter Amaelle
  • Start date
  • #1
Amaelle
310
54
Homework Statement:
look at the image
Relevant Equations:
jacobian is r^2 sinv
Greetings!
1639498591229.png


here is the solution which I undertand very well:
1639498692794.png

1639498769703.png


my question is:
if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv?

thank you!
 

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Answers and Replies

  • #2
ergospherical
881
1,208
It's sort of already baked in. The surface element comes from a little parallelogram with sides ##d\boldsymbol{l}_u = \frac{\partial \boldsymbol{\sigma}}{\partial u} du## and ##d\boldsymbol{l}_v = \frac{\partial \boldsymbol{\sigma}}{\partial v} dv## and therefore area
\begin{align*}
dS = |d\boldsymbol{l}_u \times d\boldsymbol{l}_v| = \left| \frac{\partial \boldsymbol{\sigma}}{\partial u} \times \frac{\partial \boldsymbol{\sigma}}{\partial v} \right| du dv
\end{align*}The part ##\frac{\partial \boldsymbol{\sigma}}{\partial u} \times \frac{\partial \boldsymbol{\sigma}}{\partial v}## is what your solutions denoted by the vector ##\mathbf{N}##.
 
Last edited:
  • #3
Amaelle
310
54
thank you so much!
 

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