Jacobians of 2-space to 3-space Transformation

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In summary: Eg:\frac{\partial (x,y,z)}{\partial (u,v,w)} \hbox { for 3-space and }\frac{\partial (x,y)}{\partial (u,v)} \hbox{ for 2-space }But for surface area of 3-space in the following example where the vector value function:\vec {r} = u\hat{x} + u cos(v) \hat{y} + u sin(v) \hat{z}
  • #1
yungman
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From more than one textbook, they only talking about change of variables from 2-space to 2-space or from 3-space to 3-spare. Eg:

[tex]\frac{\partial (x,y,z)}{\partial (u,v,w)} \hbox { for 3-space and }\frac{\partial (x,y)}{\partial (u,v)} \hbox{ for 2-space }[/tex]

But for surface area of 3-space in the following example where the vector value function:

[tex]\vec {r} = u\hat{x} + u cos(v) \hat{y} + u sin(v) \hat{z} [/tex]

You can see this is like:

[tex]\frac{\partial (x,y,z)}{\partial (u,v)}[/tex]

Which I don’t see this from the book. My question is whether the Jacobian is still:

[tex]| \frac{\pratial \vec{r}}{\partial u} X \frac{\pratial \vec{r}}{\partial v}| [/tex] ?

This is the standard way of finding surface area of a 3-space object. But this is like transform from 2 space [tex](u,v)[/itex] to 3-space[itex](x,y,z)[/itex].
 
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  • #2
Anyone?

I know the Jacobian is defined as:

[tex] |\frac{\partial \vec{r} }{\partial u} X \frac{\partial \vec{r} }{\partial v}|[/tex]

So it is the Jacobian even if [itex]\vec{r} = x(u,v)\hat{x} +y(u,v)\hat{y} + z(u,v)\hat{z}[/itex]

It works for finding surface intergrals. I just want to verify this here.
 
  • #3
http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant"

The domain and codomain can be of different (finite) dimensions. So in the case you're curious about, you will have
[tex] J = \frac{\partial(x,y,z)}{\partial(u,v)} [/tex]
 
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  • #4
fluxions said:
http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant"

The domain and codomain can be of different (finite) dimensions. So in the case you're curious about, you will have
[tex] J = \frac{\partial(x,y,z)}{\partial(u,v)} [/tex]

Thanks for the reply.

I am surprised Wikipedia has the answer! I gone through a lot of books and online stuff, they all only talked about square matrix where either it is 2X2 or 3X3!

That's what I suspect, because this 3X2 Jacobian work just as well in every single case.

Thanks

Alan
 
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FAQ: Jacobians of 2-space to 3-space Transformation

1. What is a Jacobian in the context of 2-space to 3-space transformation?

A Jacobian is a matrix of partial derivatives used to represent the linear transformation between two coordinate systems. In the context of 2-space to 3-space transformation, the Jacobian represents the change of variables from a two-dimensional space to a three-dimensional space.

2. How is the Jacobian calculated for a 2-space to 3-space transformation?

The Jacobian is calculated by taking the partial derivatives of each component of the transformation with respect to each variable in the original coordinate system. These partial derivatives are then arranged in a matrix, with the resulting matrix being the Jacobian.

3. What is the significance of the Jacobian in 2-space to 3-space transformation?

The Jacobian not only represents the linear transformation between two coordinate systems, but it also provides important information about the change in scale and orientation of the coordinates. It is also used in solving integrals and differential equations in multiple dimensions.

4. How does the Jacobian affect the shape of a 2-space object after transformation to 3-space?

The Jacobian affects the shape of a 2-space object after transformation to 3-space by determining how much the object is stretched or compressed in each direction. This is because the Jacobian represents the change in scale of the coordinates in the transformation.

5. Can the Jacobian be used to transform from 3-space to 2-space?

Yes, the Jacobian can be used to transform from 3-space to 2-space by taking the inverse of the Jacobian matrix. This is useful in situations where the inverse transformation is needed, such as in solving differential equations in multiple dimensions.

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