Vector Field associated with Stereographic Projection

In summary, the conversation discusses the use of the Jacobian matrix in calculating a vector field, with one participant questioning the accuracy of their calculation. They also mention plotting the results and seeking assistance from a skilled individual.
  • #1
RFeynman
7
0
Homework Statement
Calculate the field $$(\Phi_{SN})_{*} \frac{\partial}{\partial u}$$
Relevant Equations
$$(s,t) = (\Phi_{SN})(u,v) = \frac{1}{u^2 + v^2}(u,v)$$

$$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$
I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote the first equation as:

(Its also worth pointing out that $$\frac{\partial}{\partial u} = [1,0]^T$$)

$$(\Phi_{SN})_{*} \frac{\partial}{\partial u} = J_{(\Phi_{SN})} [1,0]^T$$ and after some calculations I got some gibberish, I plotted it and got this:

1617548503270.png

I think this is incorrect, in my idea the vector field should be like the contour of a circumference. Am I doing some wrong when calculating the vector field or my procedure is correct?

Thank you very much for helping!
 
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  • #2
Hey @Delta2 (as you're very skilled with vectors)I think you'd be able to answer this one; I cannot.
 
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  • #3
I am afraid I can't help with this either.
 
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Related to Vector Field associated with Stereographic Projection

1. What is a vector field associated with stereographic projection?

A vector field associated with stereographic projection is a mathematical representation of the directional changes and magnitudes of vectors on a 2-dimensional plane that is being projected onto a 3-dimensional sphere.

2. How is a vector field associated with stereographic projection calculated?

The vector field associated with stereographic projection is calculated using a mathematical formula that takes into account the coordinates of the points on the 2-dimensional plane and their corresponding points on the 3-dimensional sphere.

3. What is the significance of the vector field associated with stereographic projection?

The vector field associated with stereographic projection is significant because it helps to visualize and understand the distortions that occur when projecting a 2-dimensional plane onto a 3-dimensional sphere. It also allows for the analysis and prediction of directional changes and magnitudes of vectors on the projected surface.

4. How is the vector field associated with stereographic projection used in practical applications?

The vector field associated with stereographic projection is used in various practical applications such as navigation, meteorology, and geology. It helps to map and analyze changes in wind patterns, ocean currents, and geological features on a spherical surface.

5. Are there any limitations to the vector field associated with stereographic projection?

Yes, there are limitations to the vector field associated with stereographic projection. It is only applicable to projections onto a 3-dimensional sphere and cannot be used for other types of projections. It also does not account for the curvature of the Earth's surface, which can affect the accuracy of the projected vector field.

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