Glossary

2.2 Coordinate transformations

3 min read•august 7, 2024

Coordinate transformations are essential for switching between different ways of describing points in space. They help us move between Cartesian, cylindrical, and , each useful for specific types of problems.

These transformations involve using equations to convert coordinates and calculating Jacobian matrices. Understanding how to switch between coordinate systems is crucial for solving complex physics and engineering problems efficiently.

Coordinate Transformations

Converting between Cartesian and cylindrical coordinates

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  • Convert from (x,y,z)(x, y, z) to (r,θ,z)(r, \theta, z) using the equations:
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x})
    • z=zz = z
  • Convert from cylindrical coordinates (r,θ,z)(r, \theta, z) to Cartesian coordinates (x,y,z)(x, y, z) using the equations:
    • x=rcos⁡(θ)x = r \cos(\theta)
    • y=rsin⁡(θ)y = r \sin(\theta)
    • z=zz = z
  • Cylindrical coordinates are useful for describing systems with rotational symmetry around the zz-axis (pipes, cylinders)

Converting between cylindrical and spherical coordinates

  • Convert from cylindrical coordinates (r,θ,z)(r, \theta, z) to spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi) using the equations:
    • ρ=r2+z2\rho = \sqrt{r^2 + z^2}
    • θ=θ\theta = \theta
    • ϕ=tan⁡−1(rz)\phi = \tan^{-1}(\frac{r}{z})
  • Convert from spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi) to cylindrical coordinates (r,θ,z)(r, \theta, z) using the equations:
    • r=ρsin⁡(ϕ)r = \rho \sin(\phi)
    • θ=θ\theta = \theta
    • z=ρcos⁡(ϕ)z = \rho \cos(\phi)
  • Spherical coordinates are useful for describing systems with spherical symmetry (planets, atoms)

Converting between Cartesian and spherical coordinates

  • Convert from Cartesian coordinates (x,y,z)(x, y, z) to spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi) using the equations:
    • ρ=x2+y2+z2\rho = \sqrt{x^2 + y^2 + z^2}
    • θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x})
    • ϕ=cos⁡−1(zρ)\phi = \cos^{-1}(\frac{z}{\rho})
  • Convert from spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi) to Cartesian coordinates (x,y,z)(x, y, z) using the equations:
    • x=ρsin⁡(ϕ)cos⁡(θ)x = \rho \sin(\phi) \cos(\theta)
    • y=ρsin⁡(ϕ)sin⁡(θ)y = \rho \sin(\phi) \sin(\theta)
    • z=ρcos⁡(ϕ)z = \rho \cos(\phi)
  • Direct conversion between Cartesian and spherical coordinates is useful when a problem involves both rectangular and spherical symmetries

Transformation Techniques

Using transformation equations

  • define the relationship between coordinates in different systems
  • Apply transformation equations to convert coordinates, vector components, and other quantities between coordinate systems
  • Transformation equations preserve the physical meaning of the quantities being transformed (lengths, angles, volumes)

Calculating the Jacobian matrix

  • The J\mathbf{J} represents the partial derivatives of the new coordinates with respect to the old coordinates
  • For a transformation from coordinates (u1,u2,u3)(u_1, u_2, u_3) to (v1,v2,v3)(v_1, v_2, v_3), the Jacobian matrix is:
\frac{\partial v_1}{\partial u_1} & \frac{\partial v_1}{\partial u_2} & \frac{\partial v_1}{\partial u_3} \\ \frac{\partial v_2}{\partial u_1} & \frac{\partial v_2}{\partial u_2} & \frac{\partial v_2}{\partial u_3} \\ \frac{\partial v_3}{\partial u_1} & \frac{\partial v_3}{\partial u_2} & \frac{\partial v_3}{\partial u_3} \end{bmatrix}$$ - The [determinant of the Jacobian](https://www.fiveableKeyTerm:determinant_of_the_jacobian) matrix, $\det(\mathbf{J})$, is used to transform volume elements and integration limits between coordinate systems ### Finding inverse transformations - [Inverse transformations](https://www.fiveableKeyTerm:inverse_transformations) convert coordinates from the new system back to the original system - Obtain inverse transformations by solving the transformation equations for the original coordinates in terms of the new coordinates - Inverse transformations are useful for verifying the correctness of the original transformation and for converting results back to the original coordinate system ## Linear Algebra ### Applying change of basis - A [change of basis](https://www.fiveableKeyTerm:Change_of_Basis) transforms vectors from one set of [basis vectors](https://www.fiveableKeyTerm:basis_vectors) to another - Given a vector $\mathbf{v}$ expressed in basis $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$ and a new basis $\{\mathbf{f}_1, \mathbf{f}_2, \mathbf{f}_3\}$, the change of basis matrix $\mathbf{P}$ is: $$\mathbf{P} = \begin{bmatrix} \mathbf{f}_1 \cdot \mathbf{e}_1 & \mathbf{f}_1 \cdot \mathbf{e}_2 & \mathbf{f}_1 \cdot \mathbf{e}_3 \\ \mathbf{f}_2 \cdot \mathbf{e}_1 & \mathbf{f}_2 \cdot \mathbf{e}_2 & \mathbf{f}_2 \cdot \mathbf{e}_3 \\ \mathbf{f}_3 \cdot \mathbf{e}_1 & \mathbf{f}_3 \cdot \mathbf{e}_2 & \mathbf{f}_3 \cdot \mathbf{e}_3 \end{bmatrix}$$ - The vector $\mathbf{v}$ in the new basis is given by $\mathbf{v}_{\text{new}} = \mathbf{P}^{-1} \mathbf{v}_{\text{old}}$ - Change of basis is useful for simplifying problems by choosing a basis that aligns with the symmetries or constraints of the system (eigenbases, orthonormal bases)

Key Terms to Review (20)

Basis vectors: Basis vectors are a set of vectors in a vector space that, through linear combinations, can represent every vector in that space. They form a coordinate system that allows for the representation and manipulation of vectors across different dimensions and coordinate systems. Basis vectors are crucial when performing operations like transformations and can vary between different coordinate systems, affecting how vectors are expressed and manipulated.
Cartesian Coordinates: Cartesian coordinates are a system for defining points in a space using ordered pairs or triplets of numbers, representing distances from fixed perpendicular axes. This coordinate system is foundational in mathematics and physics, enabling the representation and manipulation of vectors, as well as facilitating analysis in various applications.
Change of Basis: Change of basis refers to the process of converting the representation of vectors or coordinates from one basis to another within a vector space. This is crucial for understanding how vector operations can be performed in different coordinate systems, allowing for consistent representation and manipulation of physical quantities in varying contexts.
Coordinate mapping: Coordinate mapping refers to the process of transforming points in one coordinate system into another, allowing for the representation of geometric figures and relationships in different contexts. This concept is fundamental in understanding how to manipulate and analyze mathematical models across varying coordinate systems, such as Cartesian, polar, or spherical coordinates. It provides the means to simplify problems by switching between systems that are more convenient for specific applications.
Cylindrical Coordinates: Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height dimension, allowing for the representation of points in space using a radius, angle, and height. This system is particularly useful for problems involving symmetry around an axis, such as those found in engineering and physics, making it easy to convert to and from Cartesian coordinates while performing vector operations, volume integrals, and coordinate transformations.
Determinant of the jacobian: The determinant of the Jacobian is a mathematical function that provides important information about the behavior of a transformation between coordinate systems. It represents the factor by which areas (in two dimensions) or volumes (in three dimensions) are scaled during this transformation, and is critical in understanding how changes in variables affect integrals in different coordinate systems.
Dimensional Consistency: Dimensional consistency refers to the principle that equations must be balanced in terms of their physical dimensions, ensuring that all terms in an equation have the same dimensional units. This concept is crucial for verifying the correctness of equations, particularly when performing coordinate transformations or solving partial differential equations, such as those encountered in heat, wave, and Laplace equations. By maintaining dimensional consistency, one can ensure that physical laws are accurately represented across different coordinate systems and mathematical formulations.
Field transformations: Field transformations refer to the mathematical operations that allow for the conversion or mapping of one field configuration into another. This concept is essential in physics, as it enables the analysis of physical systems from different perspectives or coordinate systems, thereby facilitating a deeper understanding of their behavior and properties.
Galilean Transformation: A Galilean transformation is a mathematical framework that relates the coordinates of an event as observed in one inertial reference frame to the coordinates of the same event as observed in another inertial reference frame moving at a constant velocity relative to the first. This transformation is critical for understanding how measurements of space and time change when switching between different observers moving at constant speeds, emphasizing the principles of classical mechanics and the constancy of time and space.
Inverse transformations: Inverse transformations refer to the mathematical processes that allow one to revert from a transformed coordinate system back to the original coordinate system. This concept is essential for solving problems in physics and engineering, as it helps maintain the relationships and equations consistent between different reference frames. Inverse transformations ensure that quantities and their representations remain meaningful, allowing for accurate analysis and interpretation of physical phenomena.
Jacobian Matrix: The Jacobian matrix is a matrix that represents the best linear approximation of a vector-valued function near a given point. It is composed of the first-order partial derivatives of the function, capturing how each output variable changes with respect to each input variable. This concept is crucial for understanding the behavior of vector functions and plays a significant role in coordinate transformations, where it helps relate different sets of variables in multivariable calculus.
Lorentz Transformation: The Lorentz transformation is a set of equations that relate the space and time coordinates of two observers moving at constant velocities relative to each other. This transformation is essential in the theory of relativity, as it addresses how measurements of space and time differ for observers in different inertial frames, ensuring the laws of physics are the same for all observers.
Rotation: Rotation refers to the act of turning or revolving an object around an axis or a center point. In physics and mathematics, this concept is crucial for understanding how objects move in space, especially when analyzing forces and fields in fluid dynamics and coordinate systems. It helps explain phenomena like angular momentum and is essential for transforming coordinates from one system to another, highlighting the importance of rotational symmetry in different contexts.
Scaling: Scaling refers to the process of changing the size or magnitude of a mathematical object or system, while preserving its essential properties. This concept is significant in various mathematical contexts, as it allows for the manipulation and transformation of functions, coordinates, and vector spaces, making it easier to analyze complex systems and understand their behavior under different conditions.
Shearing: Shearing refers to a type of deformation where parallel layers of material are displaced relative to each other. This concept is important in understanding how materials behave under stress, particularly in the context of coordinate transformations, where the relative position of points in space can change due to shearing effects. Shearing plays a critical role in defining how forces can result in alterations to shapes and orientations in various physical systems.
Spherical coordinates: Spherical coordinates are a three-dimensional coordinate system that represents points in space using a radius, polar angle, and azimuthal angle. This system is particularly useful for describing positions and movements in systems with spherical symmetry, making it relevant for vector operations, surface integrals, and transformations between different coordinate systems.
Trajectory analysis: Trajectory analysis is the examination of the paths that objects take through space over time, often used to understand motion dynamics and predict future positions. This analysis is critical in various fields such as physics and engineering, where it helps in interpreting data related to the movement of particles or systems. By utilizing mathematical tools and computational methods, trajectory analysis can reveal insights into behaviors and interactions of dynamic systems.
Transformation equations: Transformation equations are mathematical formulas used to relate different coordinate systems or frames of reference. They allow for the conversion of physical quantities, such as position or velocity, between these systems, enabling a clearer understanding of how observations change depending on the viewpoint. These equations are essential in fields like physics and engineering, as they help simplify complex problems by making them more manageable through appropriate coordinate choices.
Translation: Translation refers to the mathematical process of shifting a geometric figure from one position to another in a coordinate system without altering its shape or orientation. It is a fundamental operation in coordinate transformations that allows for the representation of the same object in different locations within a given space.
Unit conversion: Unit conversion is the process of changing a measurement from one unit to another, ensuring that the value remains equivalent. This concept is crucial in various fields, including science and engineering, where different units may be used to express the same physical quantity. Understanding unit conversion enables accurate communication of measurements and calculations across different systems.
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