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Curvilinear Coordinate Systems and Mathematical Physics Concepts

Explore the use of cylindrical and spherical polar coordinates in solving problems with symmetrical properties to simplify calculations. Understand concepts like gradient, divergence, curl, Laplacian, and integrals, utilizing appropriate coordinate systems. Delve into the Dirac Delta function principles and the Helmholz Theorem for vector field analysis.

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Curvilinear Coordinate Systems and Mathematical Physics Concepts

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  1. 1.4 Curvilinear Coordinates Cylindrical coordinates:

  2. Spherical polar coordinates:

  3. Gradient, Divergence, Curl, Laplacian, and Integrals For problems with spherical or cylindrical symmetry the appropriate coordinates often lead to considerable simplifications. -Take into account the derivatives of the basis vectors. -Working it out becomes often tedious. -Find the results in the textbook. -See Appendix for a compact form that generalizes to other curvilinear coordinates. -Expressions for the volume and surface elements in integrals can be found in the Appendix.

  4. 1.5 The Dirac Delta Function At the end, the delta function will appear under an integral.

  5. Consider it as the limit of an infinite thin spike of area 1. The shape of the spike does not matter.

  6. three-dimensional delta function:

  7. The integrations over the delta function can be restricted to a narrow region enclosing the spike.

  8. 1.6 Helmholz Theorem Any vector field that disappears at infinity can be expressed in terms an irrotational and a solenoidal field, which are the gradient of the scalar potential and the curl of the vector potential, respectively.

  9. Divergence-less (solenoidal) fields: curl-less (irrotational) fields:

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