1.8 KiB
Integration
Theorem: for D \subseteq \mathbb{R}^n (n=2 for simplicity) with D = X \times Y, let f: D \to \mathbb{R} then we have
\iint_D f = \int_X \Big(\int_Y f(x,y)dy \Big)dx = \int_Y \Big(\int_X f(x,y)dx \Big)dy
implying that order can be interchanged, this is true for n \in \mathbb{N}.
Proof:
will be added later.
Iteration of integrals
Theorem: for D \subseteq \mathbb{R}^n (n=2 for simplicity) bounded and piecewise smooth boundary, let f: D \to \mathbb{R} be bounded and continuous. Let R be a rectangle with D \subseteq R then
\iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad &\mathbf{x} \in D, \ 0 \quad &\mathbf{x} \notin D. \end{cases}
Proof:
will be added later.
Coordinate transformation for integrals
Theorem: for D \subseteq \mathbb{R}^n (n=2 for simplicity) bounded and piecewise smooth boundary, let f: D \to \mathbb{R} be bounded and continuous and let \phi: D \to \mathbb{R}^n be continuously differentiable and injective, define
E := \phi(D),
then we have
\iint_D f = \iint_E f \circ \phi ;\Big|\mathrm{det} \big(D_\phi \big) \Big|,
with D_\phi the Jacobian of \phi.
Proof:
will be added later.
Example
Let D = \big\{(x,y) \in \mathbb{R}^2 \;\big|\; x^2 + y^2 \leq 4 \land 0 \leq y \leq x \big\} and let \phi: D \to \mathbb{R}^2 be given by
\phi(r,\theta) = \begin{pmatrix} r\cos \theta \ r\sin \theta \end{pmatrix},
define E := \phi(D) = [0,2] \times [0, \frac{\pi}{4}]. Then E is a rectangle which can be more easily integrated.