Div and curl in other coordinate systems

In summary, the discussion is about the notation for divergence and curl in different coordinate systems. In Cartesian coordinates, these definitions can be derived from the divergence and Stokes' theorems, and are equivalent to the del operator. However, in curvilinear coordinate systems, the del operator must be applied differently, resulting in additional terms when calculating divergence and curl. The question is raised as to why the del operator is still used in these cases, and if there is a deeper understanding of how it works.
  • #1
Jezza
37
0
My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are:

[tex]\mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}[/tex]

Where [itex]S[/itex] is the closed surface bounding [itex]V[/itex], and [itex]V[/itex] is a volume whose limit contains the point [itex]P[/itex].

[tex]\left(\mathrm{curl } \vec{E}\right) \cdot \vec{n} \bigg|_P= \lim_{\Delta S \to 0} \frac{1}{\Delta S} \iint_{\ell} \vec{E} \cdot \mathrm{d} \vec{\ell}[/tex].

Where [itex]\vec n[/itex] is the unit normal to the closed contour [itex]\ell[/itex] which bounds the surface [itex]S[/itex] whose limit surrounds the point [itex]P[/itex].I also know that in cartesian coordinates, when we define the del operator [itex]\vec \nabla[/itex], these definitions are equivalent to: (Including [itex] \mathrm{grad} \phi[/itex] where [itex]\phi[/itex] is a scalar field.)
[tex]
\begin{align}
&&\vec \nabla &= \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \nonumber \\ \nonumber \\

\mathrm{grad} \phi &=& \vec \nabla \phi &= \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right) \nonumber \\ \nonumber \\

\mathrm{div } \vec{E} &=& \vec\nabla \cdot \vec E &= \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} \nonumber \\ \nonumber \\

\mathrm{curl } \vec{E} &=& \vec\nabla \times \vec E &= \left(\frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z}, \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x}, \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y}\right) \nonumber
\end{align}
[/tex]
In this case, the right hand sides can be reached from the left hand sides simply by applying vector methods, which (at least as I was led to believe) is the beauty of the [itex]\vec \nabla[/itex] notation. I'm also perfectly happy and understand that the definition of [itex]\vec \nabla[/itex] changes with coordinate system. For example, in spherical polar coordinates [itex](r, \theta, \varphi)[/itex]:
[tex]
\vec \nabla = \left(\frac{\partial}{\partial r}, \frac{1}{r} \frac{\partial}{\partial \theta}, \frac{1}{r \sin\theta} \frac{\partial}{\partial \varphi}\right)
[/tex]
The problem as I see it comes when you try and work out div, grad and curl in other coordinate systems. grad can be reached using standard vector methods, but the divergence in spherical polars is:

[tex]
\mathrm{div } \vec{E} = \frac{1}{r^2} \frac{\partial}{\partial r}\left(E_r r^2\right) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left(E_\theta \sin \theta \right) + \frac{1}{r \sin \theta}\frac{\partial E_\varphi}{\partial \varphi}
[/tex]

Which clearly cannot be reached simply using vector methods. What I'm trying to get at is that despite [itex]\vec \nabla \cdot \vec E \neq \mathrm{div} \vec E[/itex] (in traditional vector notation), we still write it! It's a similar story for curl.

It's a similar situation with cylindrical coordinates, so I assume the ability to blindly apply vector methods when calculating div and curl in cartesian coordinates is a special case. I suppose my question is why do we use this notation for general coordinate systems? How can we call del on its own an operator when actually there's more going on for the div and curl operations? Am I just missing something about how the del operator works?

Thanks for any help!
 
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  • #2
I think the point that you are missing is the following:
In rectangular coordinates, [itex] \vec{E} = E_x \vec{e_x} + E_y \vec{e_y} + E_z \vec{e_z}[/itex]. When you take [itex]\frac{\partial}{\partial x}(E_x \vec{e_x})[/itex], you just get [itex]\frac{\partial E_x}{\partial x}[/itex] because [itex] \vec{e_x}[/itex] is constant. However, in curvilinear coordinate systems, like spherical coordinates, you have [itex] \vec{E} = E_r \vec{e_r} + E_\theta \vec{e_\theta} + E_\phi \vec{e_\phi}[/itex]. When you take [itex]\frac{\partial}{\partial r}(E_r \vec{e_r})[/itex], you get one term which is [itex]\frac{\partial E_r}{\partial r}[/itex], but you get a second term that includes [itex]E_r \frac{\partial \vec e_r}{\partial r}[/itex], because [itex]\vec e_r[/itex] changes from point to point. Does this help?
 
  • Like
Likes Jezza
  • #3
It does help, thank you :)
 

Related to Div and curl in other coordinate systems

1. What are div and curl in other coordinate systems?

Div and curl are mathematical operators used in vector calculus to measure the amount and direction of fluid flow or electromagnetic fields. They are also known as divergence and curl, respectively, and are used to describe how a vector field changes at a specific point.

2. What are the other coordinate systems in which div and curl can be expressed?

Div and curl can be expressed in a variety of coordinate systems, including Cartesian, cylindrical, and spherical coordinates. Each coordinate system has its own set of equations for calculating div and curl.

3. Can div and curl be transformed between coordinate systems?

Yes, div and curl can be transformed between coordinate systems using a set of transformation equations. These equations involve derivatives and trigonometric functions, and can be used to convert div and curl values from one coordinate system to another.

4. How are div and curl used in real-world applications?

Div and curl have numerous practical applications in fields such as fluid dynamics, electricity and magnetism, and engineering. They are used to analyze and model complex systems, such as fluid flow in pipes or electric fields around charged particles.

5. What is the relationship between div and curl in other coordinate systems?

The relationship between div and curl in other coordinate systems can be described by the vector identity known as the generalized Stokes' theorem. This theorem relates the surface and volume integrals of div and curl in different coordinate systems, and is an important tool in solving problems involving these operators.

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