When to use the Jacobian in spherical coordinates?

In summary, the use of spherical coordinates already takes into account the jacobian term, as it is included in the surface element formula through the cross product of the two partial derivatives. This is represented by the vector denoted as ##\mathbf{N}## in the solution.
  • #1
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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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  • #2
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}##.
 
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  • #3
thank you so much!
 

1. What is the Jacobian in spherical coordinates?

The Jacobian in spherical coordinates is a mathematical concept used to transform a set of variables from one coordinate system to another. It is a matrix of partial derivatives that describes the relationship between the original coordinates and the new coordinates.

2. When is it necessary to use the Jacobian in spherical coordinates?

The Jacobian is necessary to use in spherical coordinates when performing integrals, differentiating vector fields, or solving differential equations. It is also useful in physics and engineering applications involving spherical symmetry.

3. How is the Jacobian calculated in spherical coordinates?

The Jacobian in spherical coordinates is calculated by taking the determinant of the matrix of partial derivatives of the transformation equations. The matrix is formed by taking the derivatives of the new coordinates with respect to the old coordinates.

4. Can the Jacobian be used in other coordinate systems?

Yes, the Jacobian can be used in other coordinate systems such as cylindrical or polar coordinates. It is a general mathematical concept that can be applied to any coordinate system to transform variables.

5. What are the benefits of using the Jacobian in spherical coordinates?

Using the Jacobian in spherical coordinates allows for easier integration and differentiation of functions, as well as simplifying the solution of differential equations. It also helps to describe physical phenomena with spherical symmetry, making it a useful tool in many scientific fields.

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