Lecture on Coordinate System Conversion

Introduction

• Continuation from previous lecture on coordinate systems.
• Focus on conversion formulas between different coordinate systems.

Rectangular to Cylindrical Coordinate Conversion

• Vector Representation:
  • Vector ( \vec{a} ) in rectangular coordinates: ( a_x \hat{i}, a_y \hat{j}, a_z \hat{k} )
  • Convert to cylindrical coordinates: ( a_\rho \hat{\rho}, a_\phi \hat{\phi}, a_z \hat{k} )
• Conversion Concepts:
  • No change in the z-component.
  • Changes occur in ( \rho ) and ( \phi ) directions.

Calculating Components

• ( a_\rho ):
  • Dot product with ( \hat{\rho} ) unit vector: ( \vec{a} \cdot \hat{\rho} )
  • Components: ( a_x \cdot \hat{\rho}, a_y \cdot \hat{\rho}, a_z \cdot \hat{\rho} ) (az component is zero)
• ( a_\phi ):
  • Dot product with ( \hat{\phi} ) unit vector: ( \vec{a} \cdot \hat{\phi} )
  • Components: ( a_x \cdot \hat{\phi}, a_y \cdot \hat{\phi}, a_z \cdot \hat{\phi} ) (az component is zero)

Conversion Table

• Coordinate System Setup:
  • x, y, z directions with unit vectors ( \hat{i}, \hat{j}, \hat{k} )
  • Rho and Phi perpendicular to Z, ( \hat{\rho} ) and ( \hat{\phi} ) perpendicular
• Dot Products for Conversion:
  • ( a_x \cdot \hat{\rho} = \cos \phi ), ( a_y \cdot \hat{\rho} = \sin \phi )
  • ( a_x \cdot \hat{\phi} = -\sin \phi ), ( a_y \cdot \hat{\phi} = \cos \phi )
  • Azimuth component remains zero.
• Example Conversion:
  • Vector ( \vec{b} = y\hat{i} - x\hat{j} + z\hat{k} )
  • Substitute and compute using conversion formulas.

Rectangular to Spherical Coordinate Conversion

• Basic Approach:
  • Similar to cylindrical conversion, but involves more angles and trigonometry.
• Conversion Steps:
  • Identify vector components ( B_r, B_\theta, B_\phi )
  • Use dot product approach with respective unit vectors ( \hat{r}, \hat{\theta}, \hat{\phi} )
• Coordinate System Setup:
  • Dot products: ( a_z \cdot \hat{r} = \cos \theta )
  • Projections for x, y components: ( r \sin \theta \cos \phi, r \sin \theta \sin \phi )
  • Conversion table setup as per angles.

Example Conversion

• Convert vector ( \vec{g} = xz/ya \hat{i} ) to spherical.
• Calculate each component using dot products and conversion formulas.

Conclusion

• Methods to convert between coordinate systems using dot products and trigonometric relations.
• Importance of conversion tables for efficient calculation.
• Refer to previous lectures for foundational concepts.