- 换元法
The area of a cube in orthogonal coordinate system and the spherical coordinate system.
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立体角
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Change of variables | MIT 18.02SC Multivariable Calculus, Fall 2010
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Change of variables (single integral and substitution) | Lecture 30 | Vector Calculus for Engineers
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Change of Variables and the Jacobian - Serpentine Integral
An infinitesimal area varies in different coordinate system because shapes of a unit area are distinct. In other words, different coordinate systems have varous scales.
In polar coordinate system:
$$ x = r⋅cosθ \\ y = r⋅sinθ $$
Jacobian matrix is a function that depicts how much each dimension of the source space should be scaled to align with another coordinate system at an arbitrary position.
The scale factor is the ratio between a target dimension and a source dimension. Thus, by multiplying those factors, the source space will be scaled to the target space.
$$ 𝐉 = \begin{bmatrix} \frac{dx}{dr} & \frac{dx}{dθ} \\ \\ \frac{dy}{dr} & \frac{dy}{dθ} \end{bmatrix} $$
Because an infinitesimal area $dx⋅dy$ only has magnitude (without direction), the area scaling factor should be just a positive real number.
To make the area in the source space to be the same in the target space, the scaling factor should be the ratio of two unit areas.
$$ \frac{d(x,y)}{d(r,θ)} $$
That is $f(x,y) = g(r,θ) dx dy$, so the integrated areas equals:
$$ ∬_D f(x,y) dx dy = ∬_E g(r,θ) dx dy = ∬_E g(r,θ) ∂(x,y)/∂(r,θ) drdθ $$
And the calculation of an area is just the cumulative product of every dimensions.
Therefore, the abosolute value of the determinant of the Jacobian matrix is taken.
$$ |det(𝐉)| = |dx/dr * dy/dθ - dx/dθ * dy/dr | = |cosθrcosθ - rsinθ(-sinθ)| = |r| $$
(2023-11-28)
The determinant of Jacobian matrix is responsible for the infinitesimal area, not the integrand.
The new integrand is just substituting the old variable with the tranformation with new variable.
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f(x,y) -> g(rcosθ, rsinθ)
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dxdy -> r drdθ
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f(x,y) = g(rcosθ, rsinθ) * r
The change of variables theorem has two aspects: