Updated several sections.
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docs/en/mathematics
multivariable-calculus
ordinary-differential-equations
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@ -77,7 +77,7 @@ will be added later.
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### The general case
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*Theorem*: Let $f: S \to \mathbb{R}$ and $\mathbf{g}: \mathbb{R}^m \to \mathbb{R}^n$ with $m$ restrictions given by
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*Theorem*: Let $f: S \to \mathbb{R}$ and $\mathbf{g}: \mathbb{R}^m \to \mathbb{R}^n$ with $m \leq n -1$ restrictions given by
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$$
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S := \big\{\mathbf{x} \in \mathbb{R}^n \;\big|\; \mathbf{g}(\mathbf{x}) = 0 \big\} \subseteq D,
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@ -15,12 +15,12 @@ will be added later.
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</details>
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<br>
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## Iteration of integralss
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## Iteration of integrals
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*Theorem*: for $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) bounded and piecewise smooth boundary, let $f: D \to \mathbb{R}$ be bounded and continuous. Let $R$ be a rectangle with $D \subseteq R$ then
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$$
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\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}
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\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}
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$$
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<details>
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@ -118,7 +118,7 @@ named the Wronskian and we can solve for $c_1(t)$ and $c_2(t)$ by integration.
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#### Ansatz method
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Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \mathbb{R}$
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Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n$ and $P_n$ polynomials of degree $\leq n$ and $\alpha \in \mathbb{R}$
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| Inhomogeneity | Particular solution |
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| ------ | --------------- |
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@ -72,4 +72,4 @@ $$
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\end{align*}
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$$
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Demanding that: $Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)$. Then $\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)$ is nonsingular. Then solve for $\mathbf{c}(t)$.
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Demanding that: $Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)$ is the Wronskian. Then $\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)$ is nonsingular. Then solve for $\mathbf{c}(t)$.
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