Is the maximum of a function in 3 space a constant?

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SUMMARY

The maximum value of the function A(R, θ, φ) = R*sin(θ)*cos(φ) in three-dimensional space is indeed a constant when evaluated over a closed volume. This maximum, denoted as A_max, represents a single value that occurs on the surface of a sphere of radius R. It is crucial to distinguish this from local maxima, which can vary across different points in the domain.

PREREQUISITES
  • Understanding of three-dimensional functions
  • Familiarity with spherical coordinates (R, θ, φ)
  • Knowledge of maxima and minima in calculus
  • Concept of closed volumes in mathematical analysis
NEXT STEPS
  • Study the properties of maxima in multivariable calculus
  • Explore spherical coordinate transformations in detail
  • Investigate the concept of local vs. global maxima
  • Learn about optimization techniques in closed domains
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Mathematicians, physicists, and students studying multivariable calculus or optimization in three-dimensional spaces will benefit from this discussion.

yungman
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I want to check with you guys:

If [itex]\; A(R,\theta,\phi) \;[/itex] is a function of [itex]R,\theta,\phi\;[/itex] in 3 space. [itex]A_{max}\;[/itex] is the maximum value of A in a closed volume. Is [itex]A_{max}\;[/itex] a constant? Seem like it is a constant, but I want to verify.

Thanks

Alan
 
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yungman said:
I want to check with you guys:

If [itex]\; A(R,\theta,\phi) \;[/itex] is a function of [itex]R,\theta,\phi\;[/itex] in 3 space. [itex]A_{max}\;[/itex] is the maximum value of A in a closed volume. Is [itex]A_{max}\;[/itex] a constant? Seem like it is a constant, but I want to verify.

Thanks

Alan


Let the function be A = R*sin(theta)*cos(phi)

The function is not constant but has maxima on the surface of a sphere of radius R.

Does this help?
 
Yes, the maximum of a function in any space, is a single number and so is a constant.

Note that this is not[]/b] a "local maximum" of which there may be many different values.
 

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