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b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex @@ -22,4 +22,13 @@ where $i$ is a \textit{multi-index}, so: \end{multicols} \drmvspace\rmvspace -The concept this formula uses is that we iterate through all possible partial derivatives of $f$ and assigns each a multi-index $i$ +The concept this formula uses is that we iterate through all possible partial derivatives of $f$ and assigns each a multi-index $i$. +To denote that we want to take the partial derivative $\partial_{112}$, we use $i = (2, 1, 0)$, since we take the derivative of the first variable twice, +of the second variable once and never of the third variable. +So the expression is thus now: +\mrmvspace +\begin{align*} + \frac{\partial_{112} f(y) (x_1 - y_1)^2 (x_2 - y_2)^1 (x_3 y_3)^0}{2!1!0!} = \frac{\partial_{112} f(y) (x_1 - y_1)^2 (x_2 - y_2)}{2} +\end{align*} + +\drmvspace diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex index 2cdb5ea..c23549a 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex @@ -1,2 +1,25 @@ -\newsectionNoPB +\newsection \subsection{Critical points} +\stepLabelNumber{all} +\compactdef{Critical Point} For $f: X \rightarrow \R^n$ differentiable, $x_0 \in X$ is called a \bi{critical point} of $f$ if $\nabla f(x_0) = 0$ +\shortremark As in 1 dimensional case, check edges of the interval for the critical point.\\ +% +To determine the kind of critical point, we need to determine if $H_f(x_0)$ is definite: +\drmvspace +\begin{multicols}{3} + \begin{itemize}[noitemsep] + \item positive definite $\Rightarrow x_0$ local max + \item negative definite $\Rightarrow x_0$ local min + \item indefinite $\Rightarrow x_0$ point of inflection + \end{itemize} +\end{multicols} + +\dhrmvspace +\setLabelNumber{all}{6} +\compactdef{Non-degenerate critical point} If $\det(H_f(x_0)) \neq 0$ (if $H_f(x_0)$ is semi-definite, then $\det(H_f(x_0)) = 0$, thus degenerate)\\ +To figure out if a matrix is definite, we can compute the eigenvalues. $A$ is positive (negative) definite, if and only if all eigenvalues are greater (lower) than $0$. +$A$ is indefinite if and only if it has both positive and negative eigenvalues. +$A$ is positive (negative) semi-definite if and only if all eigenvalues are greater (lower) or equal to $0$. +(Compute Eigenvalues using $\det(A - \lambda I) = 0$) + +For $2 \times 2$ matrices, we can use the following scheme: