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all its partial derivatives are continuous. +$f$ is of class $C^k$ if it is differentiable and each of its partial derivatives are in $C^{k - 1}$. +If $f \in C^k(X; \R^m)$ for all $k \geq 1$, then $f \in C^\infty(X; \R^m)$\\ +% ──────────────────────────────────────────────────────────────────── +\setLabelNumber{all}{4} +\compactproposition{Mixed derivatives commute} $\partial_{x, y} f = \partial_{y, x}$, as well as $\partial_{x, y, z} = \partial_{x, z, y} = \ldots$, etc (all mixed derivatives commute) +Since we have symmetry, we can use the notation $\partial_{x_1^{m_1}, \ldots, x_n^{m_n}} f = \frac{\partial^k}{\partial x^m} f = D^m f = \partial^m f$, +where $m = (m_1, \ldots, m_n)$ and $m_1 + \ldots + m_n = k$. +There are ${n + k - 1 \choose k}$ possible values for $m$ and e.g. $(1, 1, 2)$ corresponds to the derivative $\frac{\partial^4 f}{\partial x \partial y \partial^2 z}$\\ +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} +\shortremark Due to linearity of the partial derivative $\partial_x^m(a f_1 + b f_2) = a \partial_x^m f_1 + b \partial_x^m f_2$\\ +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} +\compactex{Laplace operator} $f \in C^2(X)$, $\nabla f \in C_1(X; \R^n)$, so +$\displaystyle \text{div}(\nabla f) = \sum_{i = 1}^{n} \frac{\partial}{\partial_{x_i}} \left( \frac{\partial f}{\partial_{x_i}} \right) + = \sum_{i = 1}^{n} \frac{\partial^2 f}{\partial x^2_i}$ (called \bi{Laplacian}, $\Delta f$)\\ +% ──────────────────────────────────────────────────────────────────── +\compactdef{Hessian} $f : X \rightarrow \R$ in $C^2$. For $x \in X$, the \bi{Hessian matrix} of $f$ at $x$ is the symmetric square matrix +\vspace{-0.75pc} +\begin{align*} + \text{Hess}_f(x) = (\partial_{x_i, x_j} f)_{1 \leq i, j \leq n} = H_f(x) \mediumhspace (\text{$i$-th row, $j$-th column}) +\end{align*} + +\drmvspace diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex index 68a07b7..080aa11 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex @@ -1,2 +1,5 @@ \newsectionNoPB \subsection{Change of variable} +The idea is to substitute variables for others that make the equation easier to solve. +A common example is to switch to polar coordinates from cartesian coordinates, as already demonstrated with continuity checks +% TODO: Add notes from TA notes diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex index d79479f..5fe339f 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex @@ -1,2 +1,24 @@ \newsectionNoPB \subsection{Taylor polynomials} +\compactdef{Taylor polynomials} +Let $f : X \rightarrow \R$ with $f \in C^k(X, \R)$ and $y \in X$. The Taylor-Polynomial of order $k$ of $f$ at $y$ is: +\vspace{-0.75pc} +\begin{align*} + T_k f(y; x - y) = \sum_{|i| \leq k} \frac{\partial_i f(y)(x - y)^i}{i!} +\end{align*} + +\drmvspace\rmvspace +% TODO: Find out what the \partial_1 notation means (likely TA notes 09) +where $i$ is a \textit{multi-index}, so: +\drmvspace +\begin{multicols}{2} + \begin{itemize}[noitemsep] + \item $i = (i_1, \ldots, i_n)$ (each $i_j \geq 0$) + \item $|i| = i_1 + \ldots + i_n$, $\partial_i = \partial_1^{i_1} \ldots \partial_n^{i_n}$ + \item $(x - y)^i = (x_1 - y_1)^{i_1} \cdot \ldots \cdot (x_n - y_n)^{i_n}$ + \item $i! = i_1! \cdot \ldots \cdot i_n!$ + \end{itemize} +\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 $i$