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Vector Operators: Gradient, Divergence, Curl, and General Curvilinear Coordinates

Learn the key concepts of vector calculus and how to apply the gradient, divergence, and curl operators. Understand their role in scalar and vector fields, as well as their application in polar and general curvilinear coordinates. Illustrated examples and explanations provided.

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Vector Operators: Gradient, Divergence, Curl, and General Curvilinear Coordinates

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  1. Mat-FMarch 14, 2005Vector Calculus, 10.6-10.10 Åke Nordlund Niels Obers, Sigfus Johnsen Kristoffer Hauskov Andersen Peter Browne Rønne

  2. 10: Vector operators • Why? • Because most laws of physics derived this way • conservation laws • electrodynamics … • How? • Generalization of chain rule for derivatives • Lots of examples • on the black board!

  3. Overview • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  4. Black board • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  5. Black board • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  6. Black board • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  7. Black board • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  8. Black board • Sections 10.6-10.10 • 10.6 Scalar and vector fields • scalar or vector that varies in space • 10.7 Vector operators • gradient, divergence, curl • 10.8 Vector operator formulae • learn the magic! • 10.9 Vector operators in polar coordinates • explains what was used before • 10.10 General curvilinear coordinates • generalizes e.g. polars; Einsteins theory of general rel.

  9. Enough for today! Good luck with the Exercises!

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