Notes - Continuous Mathematics HT23, Derivatives


Flashcards

Can you state the chain rule in terms of functions $f$ and $g$?


\[\frac{\text{d}\space}{\text{d}x} (g \circ f) = \left(\frac{\text{d}g}{\text{d}x} \circ f \right) \frac{\text{d}f}{\text{d}x}\]

What is

\[\frac{\text{d}\space}{\text{d}x} \left(\frac{f}{g^k}\right)\]

?


\[\frac{g \frac{\text{d}f}{\text{d}x} -kf\frac{\text{d}g}{\text{d}x}\space}{g^{k+1}\space}\]

What is a scalar field?


$f : D \to \mathbb{R}$ where $D \subseteq \mathbb{R}^n$.

What is a vector field?


$\pmb f : \mathbb{R}^n \to \mathbb{R}^m$

How is

\[\frac{\text{d}f}{\text{d}\pmb{x}\space}\]

defined where $x = [x _ 1, \ldots, x _ n]^\intercal$?


\[\left(\begin{matrix} \frac{\partial f}{\partial x_1} \\\\ \vdots \\\\ \frac{\partial f}{\partial x_i} \end{matrix}\right)\]

What is

\[\frac{d}{d\pmb{x}\space} (\pmb{x}^\intercal \pmb{a})\]

?


\[\pmb{a}\]

What is

\[\frac{d}{d\pmb{x}\space} (\pmb{x}^\intercal A \pmb{x})\]

?


\[(A + A^\intercal) \pmb{x}\]

Can you state the chain rule for

\[\frac{\text{d}\space}{\text{d}\pmb x}(g \circ f)\]

where

\[\begin{aligned} f &: \mathbb{R}^n \to \mathbb{R} \\\\ g &: \mathbb{R} \to \mathbb{R} \end{aligned}\]

?


\[\left( \frac{\text d g}{\text d x} \circ f \right) \frac{\text d f}{\text d \pmb x}\]

Can you state the chain rule for

\[\frac{\text{d}\space}{\text{d}\pmb x}(g \circ \pmb f)\]

where

\[\begin{aligned} f &: \mathbb{R}^m \to \mathbb{R}^n \\\\ g &: \mathbb{R}^n \to \mathbb{R} \end{aligned}\]

?


\[\pmb{\text J}(\pmb f)^\intercal \left( \frac{\text d g}{\text d \pmb x} \circ \pmb f \right)\]

If $f$ is a function of $x _ 1, \ldots, x _ n$, what does the Hessian look like?


\[\pmb{\text H}(f)= \begin{pmatrix}\frac{\partial^2f}{\partial x_1 \partial x_1} & \cdots & \frac{\partial^2f}{\partial x_1 \partial x_n} \\\\ \vdots & \ddots & \vdots \\\\ \frac{\partial^2f}{\partial x_n \partial x_1} & \cdots & \frac{\partial^2f}{\partial x_n \partial x_n}\end{pmatrix}\]

When will the Hessian matrix of $f$ be symmetric?


When $f$ is continuous.

If $f$ is a function of $x _ 1, \ldots, x _ n$, then what is

\[\pmb{\text H}(f)_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}\]

If a vector-valued function $\pmb f(\pmb x)$ is equal to

\[\pmb f(\pmb x) = \begin{pmatrix}f_1(\pmb x) \\\\\vdots\\\\f_m(\pmb x)\end{pmatrix}\]

where $\pmb x = [x _ 1, \ldots, x _ n]^\intercal$ then what is the Jacobian of $\pmb f$?


\[\pmb{\text J}(\pmb f) = \begin{pmatrix}\frac{\partial f_1}{\partial x_1 } & \cdots & \frac{\partial f_1}{\partial x_n} \\\\ \vdots & \ddots & \vdots \\\\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n}\end{pmatrix}\]

If a vector-valued function $\pmb f(\pmb x)$ is equal to

\[\pmb f(\pmb x) = \begin{pmatrix}f_1(\pmb x) \\\\\vdots\\\\f_m(\pmb x)\end{pmatrix}\]

then what does the $i$-th (from $1 < i < m$) row of the Jacobian $\pmb{\text J}$ give you?


The gradient of $f _ i$ .

Let $f : \mathbb{R}^n \to \mathbb{R}$. What is the Hessian matrix $\pmb{\text H}(f)$ in terms of the Jacobian?


\[\pmb{\text H}(f) = \pmb{\text J}\left(\frac{\text d f}{\text{d}\pmb x}\right)^\intercal\]

What is

\[\pmb{\text J}(\pmb f + \pmb g)\]

?


\[\pmb{\text J}(\pmb f) + \pmb{\text J}(\pmb g)\]

What is

\[\pmb{\text J} (c \pmb f)\]

?


\[c\pmb{\text J}(\pmb f)\]

What is

\[\pmb{\text J}(A\pmb f)\]

where $A$ is a constant matrix?


\[A\pmb{\text J}(\pmb f)\]

What is

\[\pmb{\text J}(\pmb f^\intercal \pmb g)\]

?


\[\pmb g^\intercal \pmb{\text J}(\pmb f) + f^\intercal \pmb{\text J}(\pmb g)\]

What is

\[\pmb{\text J}(f \pmb g)\]

where $f : \mathbb{R}^m \to \mathbb{R}$ ?


\[\pmb g \frac{\text d f}{\text d x}^\intercal + f \pmb{\text J}(\pmb g)\]

What is

\[\pmb{\text J}(\pmb g \circ \pmb f)\]

(this is the chain rule not the product rule, actually read)?


\[(\pmb{\text J}(\pmb g) \circ \pmb f)\pmb{\text J}(\pmb f)\]

What is

\[\pmb{\text J}(\pmb x)\]

?


\[I\]

What is

\[\pmb{\text J}(A\pmb x)\]

where $A$ is a constant matrix?


\[A\]

What is

\[\pmb{\text J}(f(\pmb x)\pmb c)\]

?


\[\pmb c \frac{\text d f}{\text d x}^\intercal\]

What is always true about the dimensions of the Hessian in comparison to the dimension of the Jacobian?


The Hessian is always square, the Jacobian not necessarily so.




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