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Updated several sections.

This commit is contained in:
Luc Bijl 2023-11-02 09:48:51 +01:00
parent b46e2fd38e
commit c5aeff38b9
4 changed files with 5 additions and 5 deletions
docs/en/mathematics
multivariable-calculus
ordinary-differential-equations

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@ -77,7 +77,7 @@ will be added later.
### The general case
*Theorem*: Let $f: S \to \mathbb{R}$ and $\mathbf{g}: \mathbb{R}^m \to \mathbb{R}^n$ with $m$ restrictions given by
*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
$$
S := \big\{\mathbf{x} \in \mathbb{R}^n \;\big|\; \mathbf{g}(\mathbf{x}) = 0 \big\} \subseteq D,

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</details>
<br>
## Iteration of integralss
## 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
$$
\iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad \mathbf{x} \in D, \\ 0 \quad \mathbf{x} \notin D. \end{cases}
\iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad &\mathbf{x} \in D, \\ 0 \quad &\mathbf{x} \notin D. \end{cases}
$$
<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.
#### Ansatz method
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 $\alpha \in \mathbb{R}$
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 $\alpha \in \mathbb{R}$
| Inhomogeneity | Particular solution |
| ------ | --------------- |

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\end{align*}
$$
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 $\mathbf{c}(t)$.
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 $\mathbf{c}(t)$.