Calculus: Double Integrals and Change of Variables

Key vocabulary and mathematical definitions from the lecture on double integrals, coordinate transformations, and polar coordinates.

Iterated Integral (累次積分)

A method of calculating a double integral by performing one-variable integration repeatedly, one variable at a time.

Rectangle Region (長方形領域)

An integration region $D$ where the limits of integration for $x$ and $y$ are independent of each other, typically defined as a \req x \req b and c \req y \req d.

Separable Form (変数分離形)

A property of a function $f(x, y)$ when it can be expressed as a product of two single-variable functions, such that $f(x, y) = f_1(x) f_2(y)$.

Regular Linear Transformation (正則な一次変換)

A transformation of the form $x = au + bv$ and $y = cu + dv$ where the determinant of the matrix of coefficients, $\text{ad} - \text{bc}$, is not equal to zero.

Jacobian Matrix (Jacobian行列)

The matrix $J$ of first-order partial derivatives for a transformation $x = g(u, v)$ and $y = h(u, v)$, expressed as $J = \begin{pmatrix} g_u & g_v \\ h_u & h_v \end{pmatrix}$.

Jacobian (Jacobian / ヤコビアン)

The determinant of the Jacobian matrix, denoted as $\det J = g_u h_v - g_v h_u$, which represents the rate of change of area during a coordinate transformation.

Change of Variables Formula

The formula for double integrals stating that $\iint_D f(x, y) dx dy = \iint_E f(g(u, v), h(u, v)) | \det J | du dv$.

Area Transformation Property

When $f(x, y) = 1$, the relationship where the area of region $D$ equals the area of region $E$ multiplied by the absolute value of the Jacobian, or $\iint_D 1 dx dy = | \det J | \iint_E 1 du dv$.

Polar Coordinates (極座標)

A coordinate system where a point $(x, y)$ is defined by a radius $r$ and an angle $\theta$, with the conversion formulas $x = r \cos(\theta)$ and $y = r \sin(\theta)$, where $r = \sqrt{x^2 + y^2}$.

Jacobian for Polar Coordinates

The specific Jacobian value for the transformation to polar coordinates, which is derived as $\det J = r$.

Gaussian Integral

An integral resulting in the value $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$ , which is proven in the lecture using double integration and the polar coordinate transformation.