Consider the polar to cartesian coordinate transformation
𝑥
=
𝑟
cos
⁡
𝜃
and
𝑦
=
𝑟
sin
⁡
𝜃
the Jacobian for this would be

1)
𝑟

2)
𝑟
2

3)
𝑟
2
sin
⁡
𝜃

4)
𝑟
sin
⁡
𝜃
cos
⁡
𝜃

Question

Consider the polar to cartesian coordinate transformation 
𝑥
=
𝑟
cos
⁡
𝜃
 and 
𝑦
=
𝑟
sin
⁡
𝜃
 the Jacobian for this would be

1) 
𝑟

2) 
𝑟
2

3) 
𝑟
2
sin
⁡
𝜃

4) 
𝑟
sin
⁡
𝜃
cos
⁡
𝜃

BlackTom AI · Accepted Answer

The Jacobian determinant for polar coordinates is computed from the matrix of partial derivatives ∂(x,y)/∂(r,θ).

1) Relevant knowledge points and concepts
- For a change of variables (r, θ) -> (x, y) with x = r cosθ, y = r sinθ, the Jacobian is det[∂(x,y)/∂(r,θ)].
- The Jacobian gives the local area-scaling factor when converting integrals: dA = |J| dr dθ.
- Trigonometric identity cos^2θ + sin^2θ = 1 will be used to simplify the determinant.

2) Complete reasoning process, step by step
- Build the Jacobian matrix of partial derivatives:
  ∂x/∂r = cosθ
  ∂x/∂θ = -r sinθ
  ∂y/∂r = sinθ
  ∂y/∂θ = r cosθ
- So the Jacobian matrix is:
  [ cosθ    -r sinθ ]
  [ sinθ     r cosθ ]

3) Calculations with intermediate steps
- Compute the determinant:
  J = (cosθ)(r cosθ) - (-r sinθ)(sinθ)
    = r cos^2θ + r sin^2θ
    = r (cos^2θ + sin^2θ)
    = r · 1
    = r
- Thus the Jacobian determinant equals r. If needed in integrals, use |J| = r (for r ≥ 0 this is just r).

4) Why the correct answer is correct
- The determinant reduces exactly to r because the r factors multiply the trig squares which sum to 1. That is the precise area-scaling factor converting dr dθ to dx dy in polar coordinates.

5) Why the other options are incorrect
- r^2 would arise if one mistakenly multiplied r from both partial derivatives in a way that double-counts r; here only one r appears after determinant expansion.
- r^2 sinθ is dimensionally and algebraically incorrect; no such extra sinθ factor remains once cos^2θ + sin^2θ is simplified.
- r sinθ cosθ corresponds to the product of off-diagonal terms without taking the determinant sign and the sum; it is not the full determinant.

6) Conclusion
- The Jacobian is r, matching option 1.

Consider the polar to cartesian coordinate transformation 𝑥 = 𝑟 cos ⁡ 𝜃 and 𝑦 = 𝑟 sin ⁡ 𝜃 the Jacobian for this would be 1) 𝑟 2) 𝑟 2 3) 𝑟 2 sin ⁡ 𝜃 4) 𝑟 sin ⁡ 𝜃 cos ⁡ 𝜃单项选择题

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