Y = ρ sin(φ) sin(θ)

Definition

The equation $y = \rho \sin(\phi) \sin(\theta)$ describes the y-coordinate in a spherical coordinate system, which represents a point in three-dimensional space. This coordinate system is defined by three parameters: the radial distance $\rho$, the polar angle $\phi$, and the azimuthal angle $\theta$. The relationship allows for conversion between Cartesian and spherical coordinates, which is essential for solving various problems in calculus and physics.

5 Must Know Facts For Your Next Test

1. In spherical coordinates, the angles $\phi$ and $\theta$ are crucial for determining the orientation of the point in space.
2. The equation shows how the y-coordinate can be derived using both angles and the radial distance, emphasizing their interdependence.
3. When $\phi$ is equal to $\frac{\pi}{2}$ (90 degrees), the equation simplifies to $y = \rho \sin(\theta)$, indicating that it lies in the xy-plane.
4. Understanding this equation is fundamental for transforming integrals and solving triple integrals in spherical coordinates.
5. Visualizing points in spherical coordinates requires grasping how $\rho$, $\phi$, and $\theta$ interact to form a point's position in 3D space.

Review Questions

• How does the equation $y = \rho \sin(\phi) \sin(\theta)$ illustrate the relationship between spherical and Cartesian coordinates?
  • The equation highlights how to convert from spherical to Cartesian coordinates by expressing the y-coordinate in terms of the radial distance $\rho$, polar angle $\phi$, and azimuthal angle $\theta$. In spherical coordinates, each parameter contributes to defining a point's location in three-dimensional space. This relationship allows us to understand how changes in these angles or distances affect a point's position along the y-axis.
• What significance does understanding the angles $\phi$ and $\theta$ have in visualizing points represented by the equation?
  • Understanding angles $\phi$ and $\theta$ is crucial because they determine how a point is oriented in 3D space. The angle $\phi$ affects vertical positioning relative to the z-axis, while $\theta$ determines horizontal rotation around that axis. By manipulating these angles, one can visualize how points move through space, providing insights into surface areas or volumes when integrating in spherical coordinates.
• Evaluate how changing the radial distance $\rho$ influences the overall position of a point defined by $y = \rho \sin(\phi) \sin(\theta)$ in spherical coordinates.
  • Changing the radial distance $\rho$ directly impacts all three Cartesian coordinates derived from spherical ones, including y. As $\rho$ increases, the point moves further away from the origin, expanding its reach throughout all dimensions based on its angular parameters. This relationship emphasizes how radial distance influences spatial positioning, which is fundamental when analyzing physical phenomena or performing transformations.

Related terms

Spherical Coordinates: A system of coordinates that uses three parameters (radial distance, polar angle, azimuthal angle) to define a point in three-dimensional space.

Radial Distance ($\rho$): The distance from the origin to the point in three-dimensional space, representing how far out from the center the point is.

Polar Angle ($\phi$): The angle measured from the positive z-axis down to the point, indicating how high or low the point is in relation to the z-axis.