Taking the time derivative of a curl

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SUMMARY

The discussion clarifies the commutativity of time derivatives and curls in vector calculus. It establishes that while the partial time derivative of a curl is commutative, the total time derivative is not due to the dependency of coordinate variables on time in Cartesian, spherical, and cylindrical coordinates. The distinction between coordinate variables and position variables is emphasized, highlighting that curls operate on fields rather than objects. The conclusion is that time derivatives and curls are commutative when applied to vector fields defined over orthogonal coordinate systems.

PREREQUISITES
  • Understanding of vector calculus concepts, particularly curls and derivatives.
  • Familiarity with Cartesian, spherical, and cylindrical coordinate systems.
  • Knowledge of the distinction between coordinate variables and position variables.
  • Basic principles of field theory in physics and mathematics.
NEXT STEPS
  • Study the properties of partial and total derivatives in vector calculus.
  • Explore the implications of coordinate transformations on vector fields.
  • Learn about the application of curls in fluid dynamics and electromagnetism.
  • Investigate the role of orthogonality in vector field analysis.
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Students and professionals in mathematics, physics, and engineering, particularly those focusing on vector calculus, fluid dynamics, and field theory.

nabeel17
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Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and cylindrical coordinates, even the unit vectors depend on time? Also the partial derivative of a curl in curvilinear coordinates is commutative?
 
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I think you are confusing coordinate variables (which are usually called x, y, z, t) with position variables (which are also usually called x, y, z, t). The difference is that coordinate variables refer to a fixed grid over which a vector field is defined, and position variables refer to some dynamic position of some object.

A field has a value at each coordinate position, whereas an object only occupies a particular subset of coordinate positions. The time derivative and curl are commutative since they are operations on a field, and the field coordinates are (typically) orthogonal. It doesn't make sense to take the curl of an object position.
 

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