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Chapter 15: Problem 12

Express the vector field \(\partial_{\varphi}\) in polar coordinates \((\theta, \phi)\) on the unit 2 -sphere in terms of stereographic coordinates \(X\) and \(Y\).

Short Answer

Expert verified
\(\partial_{\varphi} = -\frac{sin^2\theta sin\varphi}{(1 - cos\theta)^2} \partial_{X} + \frac{sin^2\theta cos\varphi}{(1 - cos\theta)^2} \partial_{Y}\)

Step by step solution

01

Define the Stereographic projection

The Stereographic projection from the unit sphere to the plane is given by the equations \(X = \frac{sin\theta cos\varphi}{1 - cos\theta}\) and \(Y = \frac{sin\theta sin\varphi}{1 - cos\theta}\).
02

Differentiate X and Y with respect to \(\theta\) and \(\varphi\)

We find \(\partial_{\varphi}X\) and \(\partial_{\varphi}Y\) by differentiating X and Y with respect to \(\varphi\). Thus, we have \(\partial_{\varphi}X = -\frac{sin^2\theta sin\varphi}{(1 - cos\theta)^2}\) and \(\partial_{\varphi}Y = \frac{sin^2\theta cos\varphi}{(1 - cos\theta)^2}\).
03

Express the vector \(\partial_{\varphi}\) in terms of X and Y

By substituting the expressions for \(\partial_{\varphi}X\) and \(\partial_{\varphi}Y\) into the vector \(\partial_{\varphi}\), we obtain \(\partial_{\varphi} = \partial_{\varphi}X \partial_{X} + \partial_{\varphi}Y \partial_{Y}\ = -\frac{sin^2\theta sin\varphi}{(1 - cos\theta)^2} \partial_{X} + \frac{sin^2\theta cos\varphi}{(1 - cos\theta)^2} \partial_{Y}\).

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Most popular questions from this chapter

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