Jacobians & Coordinate Transformations

The Jacobian describes how a transformation stretches space. Learn the Jacobian matrix and determinant, change of variables, Cartesian ↔ polar, cylindrical and spherical transformations with their Jacobians (r, r, r² sin φ), inverse transformations, and robotics and FEM applications.

Integrating over a circle is awkward in x and y but easy in r and θ — provided you include the Jacobian factor r. Jacobians also map joint velocities to end-effector velocities in robots and map element coordinates in finite element analysis.

An analogy

The Jacobian determinant is a 'stretch factor': it tells how much a tiny square in one coordinate system grows or shrinks when mapped to another.

Why a factor appears

In polar coordinates, a small patch dr × dθ has area r dr dθ — patches further from the origin are bigger. Forgetting r is the most common error.

Choosing coordinates

Polar for discs and sectors, cylindrical for pipes and shafts, spherical for spheres and radiation fields. Pick coordinates matching the geometry's symmetry.

Jacobian matrix

For vector functions, J holds all first partials; it is the best linear approximation of the mapping. det J = 0 means the mapping collapses (singularity).

Applications

Robot velocity kinematics ẋ = J(q) q̇; singular configurations where the arm loses a degree of freedom; isoparametric FEM elements map a reference square to real element shapes using |J|.

Formulas

• 2D Jacobian: |J| = ∂(x, y)/∂(u, v) = x_u y_v − x_v y_u
• Change of variables: ∬ f dx dy = ∬ f |J| du dv
• Polar / cylindrical: x = r cos θ, y = r sin θ; dA = r dr dθ, dV = r dr dθ dz
• Spherical: x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ; dV = ρ² sinφ dρ dθ dφ
• Inverse: ∂(u, v)/∂(x, y) = 1/(∂(x, y)/∂(u, v))

Worked example: Polar Jacobian

Given: x = r cos θ, y = r sin θ

Find: |J|.

• x_r = cos θ, x_θ = −r sin θ; y_r = sin θ, y_θ = r cos θ
• |J| = r cos²θ + r sin²θ = r

Answer: |J| = r

Worked example: Gaussian integral

Given: ∬_ℝ² e^(−(x² + y²)) dA

Find: Value.

• Polar: ∫₀^{2π}∫₀^∞ e^(−r²) r dr dθ
• = 2π × ½

Answer: π (so ∫e^(−x²)dx = √π)

Common mistakes

• Wrong: Omitting r or ρ² sin φ. — Right: Always include |J| in the new integral.
• Wrong: Mixing θ and φ conventions. — Right: State which angle is polar vs azimuth.

Applications

• Robotics: Velocity and force mapping, singularities.
• FEM: Isoparametric element mapping.
• EM fields & antennas: Spherical integration of radiation.

Key takeaways

• |J| = stretch factor.
• Polar r, cylindrical r, spherical ρ² sin φ.
• Match coordinates to symmetry.
• det J = 0 → singularity.

Going further

• Metric tensors: Generalised coordinate systems (M16).
• Manipulability: √det(JJᵀ) in robotics.
• Jacobian in Newton's method: Systems of non-linear equations.