Vector Operations in Polar Coordinates?

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SUMMARY

Vector operations in polar coordinates, particularly in \mathbb{R}^3, are less convenient than using rectangular coordinates. While spherical coordinates can be useful for spherically symmetrical vector fields, operations such as vector addition and dot products are not straightforward in polar coordinates. The discussion emphasizes the importance of using linear coordinate systems for vector operations, particularly in physics, where cylindrical coordinates may also be beneficial for cylindrical symmetry. Ultimately, the choice of coordinate system depends on the specific application.

PREREQUISITES
  • Understanding of vector operations in \mathbb{R}^3
  • Familiarity with spherical and cylindrical coordinate systems
  • Knowledge of vector calculus concepts such as gradient, divergence, and curl
  • Basic proficiency in physics, particularly in vector fields
NEXT STEPS
  • Research the application of spherical coordinates in vector fields
  • Learn about cylindrical coordinates and their advantages in specific scenarios
  • Study vector calculus operations like grad, div, and curl in different coordinate systems
  • Explore the limitations of polar coordinates in vector operations
USEFUL FOR

Students and professionals in physics, mathematicians, and engineers who require a deep understanding of vector operations and coordinate systems in their work.

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Do you think you could do vector operations in polar coordinates?
 
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When you're in [itex]\mathbb{R}^3[/itex] and you want to designate the tip of a vector by giving three coordinates, then you can use spherical coordinates (or polar coordinates in [itex]\mathbb{R}^2[/itex]) or any other coordinate system.

But using rectangular coordinates is much more convenient, because the vector notation in component form will mean the first component times the first basis-vector, the second component times the second basis-vector and so on. This can't be done in polar/spherical coordinates.
Also, adding vectors component-wise, and things like the dot-product are of no use either.

So much of the reason why you use vectors will be lost when going to spherical/polar coordinates.

In short, with vectors: use a linear coordinate system.
 
Depends what you're doing. In physics, it is often convenient to use spherical polar coordinates for vector fields, particular if the field is spherically symmetrical. If you have cylindrical symmetry, cylindrical polar coordinates are often useful.

You can certainly write grad f, div V, and curl V in terms of their polar components.
 

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