15.1 Spherical coordinate system and transformation

Spherical coordinates offer a powerful way to describe points in 3D space using radius, polar angle, and azimuthal angle. This system is especially useful for problems with spherical symmetry, like calculating volumes of spheres or analyzing radial fields.

Understanding the relationships between spherical and Cartesian coordinates is crucial for solving complex 3D problems. The Jacobian determinant and volume element in spherical coordinates are key tools for performing triple integrals, enabling us to tackle a wide range of real-world applications.

Spherical Coordinate System

Coordinate System Components

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• Spherical coordinates define a point in 3D space using radius, polar angle, and azimuthal angle
• Radius (ρ) measures the distance from the origin to the point
  • Always non-negative ($ρ ≥ 0$)
• Polar angle (θ) measures the angle between the positive z-axis and the line from origin to point
  • Range is $0 ≤ θ ≤ π$
  • $θ = 0$ corresponds to positive z-axis, $θ = π/2$ corresponds to xy-plane, $θ = π$ corresponds to negative z-axis
• Azimuthal angle (φ) measures the angle in the xy-plane from positive x-axis to the projection of line from origin to point
  • Range is $0 ≤ φ < 2π$
  • $φ = 0$ corresponds to positive x-axis, $φ = π/2$ to positive y-axis, $φ = π$ to negative x-axis, $φ = 3π/2$ to negative y-axis

Coordinate Relationships

• Points with same radius $ρ$ lie on a sphere centered at origin
• Points with same polar angle $θ$ lie on a circular cone with vertex at origin and axis along the z-axis
  • Angle between cone and z-axis is $θ$
• Points with same azimuthal angle $φ$ lie on a vertical half-plane emanating from z-axis
  • Half-plane makes angle $φ$ with the xz-plane

Coordinate Transformations

Cartesian to Spherical

• Given Cartesian coordinates $(x, y, z)$, spherical coordinates $(ρ, θ, φ)$ are:
  • $ρ = \sqrt{x^2 + y^2 + z^2}$
  • $θ = \arccos(\frac{z}{\sqrt{x^2 + y^2 + z^2}})$
    • If $ρ = 0$, $θ$ is undefined
  • $φ = \arctan(\frac{y}{x})$
    • If $x > 0$, $φ = \arctan(\frac{y}{x})$
    • If $x < 0$ and $y ≥ 0$, $φ = \arctan(\frac{y}{x}) + π$
    • If $x < 0$ and $y < 0$, $φ = \arctan(\frac{y}{x}) - π$
    • If $x = 0$ and $y > 0$, $φ = π/2$
    • If $x = 0$ and $y < 0$, $φ = -π/2$
    • If $x = 0$ and $y = 0$, $φ$ is undefined

Spherical to Cartesian

• Given spherical coordinates $(ρ, θ, φ)$, Cartesian coordinates $(x, y, z)$ are:
  • $x = ρ \sin θ \cos φ$
  • $y = ρ \sin θ \sin φ$
  • $z = ρ \cos θ$

Jacobian Determinant

• The Jacobian determinant $|J|$ is used for change of variables in multiple integrals
• For transformation from Cartesian $(x, y, z)$ to spherical $(ρ, θ, φ)$:

\frac{∂x}{∂ρ} & \frac{∂x}{∂θ} & \frac{∂x}{∂φ} \\ \frac{∂y}{∂ρ} & \frac{∂y}{∂θ} & \frac{∂y}{∂φ} \\ \frac{∂z}{∂ρ} & \frac{∂z}{∂θ} & \frac{∂z}{∂φ} \end{vmatrix} = ρ^2 \sin θ$$ ## Integration in Spherical Coordinates ### Volume Element - The volume element $dV$ in spherical coordinates is the Jacobian determinant $|J|$ multiplied by the differentials: $$dV = |J| dρ dθ dφ = ρ^2 \sin θ dρ dθ dφ$$ - Triple integrals in spherical coordinates have the form: $$\iiint_E f(ρ,θ,φ) ρ^2 \sin θ dρ dθ dφ$$ - $E$ is the region of integration - $f(ρ,θ,φ)$ is the integrand function in spherical coordinates - Order of integration is typically $dρ$ first, then $dθ$, then $dφ$ - Limits of integration depend on the specific region $E$