答其他回答（2条）

• 美髯公想 问答领域知识达人 2026-09-30 06:07:08

  Laplace's equation in polar coordinates, also known as the polar form of Laplace's equation, can be derived by expressing Laplace's equation in Cartesian coordinates and transforming it to polar coordinates. Here is the derivation:

  Laplace's equation in Cartesian coordinates:

  ∇²φ = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² = 0

  Transformation to polar coordinates:

  x = rcosθ

  y = rsinθ

  z = z

  Differentiating the transformation equations with respect to x, y, and z, we get:

  ∂x/∂r = cosθ

  ∂x/∂θ = -rsinθ

  ∂y/∂r = sinθ

  ∂y/∂θ = rcosθ

  ∂z/∂z = 1

  Using the chain rule, we can express the second-order derivatives of φ with respect to x, y, and z in terms of the second-order derivatives of φ with respect to r and θ:

  ∂²φ/∂x² = (∂²φ/∂r²)cos²θ - (2∂²φ/∂r∂θ)cosθsinθ + (∂²φ/∂θ²)sin²θ

  ∂²φ/∂y² = (∂²φ/∂r²)sin²θ + (2∂²φ/∂r∂θ)cosθsinθ + (∂²φ/∂θ²)cos²θ

  ∂²φ/∂z² = ∂²φ/∂z²

  Substituting these expressions into Laplace's equation in Cartesian coordinates, we have:

  0 = (∂²φ/∂r²)cos²θ - (2∂²φ/∂r∂θ)cosθsinθ + (∂²φ/∂θ²)sin²θ + (∂²φ/∂r²)sin²θ + (2∂²φ/∂r∂θ)cosθsinθ + (∂²φ/∂θ²)cos²θ + ∂²φ/∂z²

  Simplifying the equation by canceling out terms, we obtain:

  0 = (∂²φ/∂r²) + (1)(∂φ/∂r) + (1²)(∂²φ/∂θ²) + (∂²φ/∂z²)

  Finally, we can express Laplace's equation in polar coordinates (r, θ, z) as:

  ∇²φ = (∂²φ/∂r²) + (1)(∂φ/∂r) + (1²)(∂²φ/∂θ²) + (∂²φ/∂z²) = 0

  This is the polar form of Laplace's equation, where the Laplacian operator is expressed in terms of the second derivatives with respect to r, θ, and z.

• 蜗壳篮球 问答领域知识达人 2026-09-30 06:07:08

  直角坐标下的拉普拉斯方程为：(ə²/əx²)+(ə²/əy²)f=0

  极坐标下的拉普拉斯方程：(ə²/ər²)+(1)(ə/ər)+(1²)(ə²/ə²θ)f=0

  下面的极坐标下的拉普拉斯方程是怎么推导出的呢？

  f是函数，ə是求偏导符号

  直角坐标下的拉普拉斯方程为：(ə²/əx²)+(ə²/əy²)f=0

  极坐标下的拉普拉斯方程：(ə²/ər²)+(1)(ə/ər)+(1²)(ə²/ə²θ)f=0