Notes - Continuous Mathematics HT23, Derivatives
Flashcards
Can you state the chain rule in terms of functions $f$ and $g$?
What is
\[\frac{\text{d}\space}{\text{d}x} \left(\frac{f}{g^k}\right)\]
?
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 $x = [x _ 1, \ldots, x _ n]^\intercal$?
What is
\[\frac{d}{d\pmb{x}\space} (\pmb{x}^\intercal \pmb{a})\]
?
What is
\[\frac{d}{d\pmb{x}\space} (\pmb{x}^\intercal A \pmb{x})\]
?
Can you state the chain rule for
\[\frac{\text{d}\space}{\text{d}\pmb x}(g \circ f)\]
where
\[\begin{aligned}
f &: \mathbb{R}^n \to \mathbb{R} \\\\
g &: \mathbb{R} \to \mathbb{R}
\end{aligned}\]
?
Can you state the chain rule for
\[\frac{\text{d}\space}{\text{d}\pmb x}(g \circ \pmb f)\]
where
\[\begin{aligned}
f &: \mathbb{R}^m \to \mathbb{R}^n \\\\
g &: \mathbb{R}^n \to \mathbb{R}
\end{aligned}\]
?
If $f$ is a function of $x _ 1, \ldots, x _ n$, what does the Hessian look like?
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 $\pmb f$?
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?
What is
\[\pmb{\text J}(\pmb f + \pmb g)\]
?
What is
\[\pmb{\text J} (c \pmb f)\]
?
What is
\[\pmb{\text J}(A\pmb f)\]
where $A$ is a constant matrix?
What is
\[\pmb{\text J}(\pmb f^\intercal \pmb g)\]
?
What is
\[\pmb{\text J}(f \pmb g)\]
where $f : \mathbb{R}^m \to \mathbb{R}$ ?
What is
\[\pmb{\text J}(\pmb g \circ \pmb f)\]
(this is the chain rule not the product rule, actually read)?
What is
\[\pmb{\text J}(\pmb x)\]
?
What is
\[\pmb{\text J}(A\pmb x)\]
where $A$ is a constant matrix?
What is
\[\pmb{\text J}(f(\pmb x)\pmb c)\]
?
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.