Gradient in Spherical Coordinates: Computing w/ {em} & {wm}

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SUMMARY

The discussion focuses on computing the gradient of a scalar function \(\phi\) in spherical coordinates using the equation \(G-1d\phi\). The user seeks clarification on how to express this gradient in terms of the energy-momentum tensor ({em}) and the wave-momentum tensor ({wm}). It is essential to understand the conversion of co-vectors and vectors into scalars and the use of charts to define the gradient as a vector field on the sphere, pulling it back from \(\mathbb{R}^n\).

PREREQUISITES
  • Spherical coordinates and their properties
  • Gradient computation in vector calculus
  • Understanding of co-vectors and vectors
  • Charts and manifolds in differential geometry
NEXT STEPS
  • Study the computation of gradients in spherical coordinates
  • Learn about the relationship between vectors and co-vectors
  • Explore the use of charts in defining vector fields
  • Investigate the applications of the energy-momentum tensor ({em}) and wave-momentum tensor ({wm}) in physics
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Mathematicians, physicists, and students studying differential geometry or vector calculus, particularly those working with spherical coordinates and tensor analysis.

autobot.d
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So I am working in spherical coordinates and to find the gradient I have the eqn

G-1d\phi
where \phi is a scalar function

Then I am supposed to compute in terms of {em} and {wm}.

I am just confused what it means to compute in terms of? Do i have to convert the
co and contra vectors into scalars? Any help is appreciated.
 
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Just confused on what the gradient of phi in terms of the vector and covector in spherical coordinates are. Maybe that is a little easier to help with?
 
You need to use charts to define the gradient --seen as a vector field--on the sphere.

Use the charts to pull it back from R^n.
 

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