Taylor expansion

Main Question or Discussion Point

I don't understand this as isn't according to chain rule,
.
So where is the
in the above derivative of F(t)?
Source: http://www.math.ubc.ca/~feldman/m226/taylor2d.pdf

blue_leaf77
Homework Helper
$F(t) = f(x(t),y(t))$, so despite written to be a function of $t$, the functional form of $F(t)$ is not written explicitly as a function of $t$.

$F(t) = f(x(t),y(t))$, so despite written to be a function of $t$, the functional form of $F(t)$ is not written explicitly as a function of $t$.
Why is it not a function of t? I am still new to this so do you have something i can read on about this?

blue_leaf77
Homework Helper
Why is it not a function of t?
It's not an explicit function of $t$. When you want to do chain rule, you have to pay attention on which variables are written explicitly, despite whether or not these variables are functions of yet another variable. For example take $F(t) = xy^2$ where $x = \sqrt{t}$ and $y=t-2$. If you want to calculate $dF/dt$, you can either first express $x$ and $y$ in terms of $t$ and then differentiate w.r.t. $t$ or let $F$ be expressed in $x$ and $y$ then use the chain rule
$$\frac{dF}{dt} = \frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt} .$$
I guess this problem should belong to multivariate calculus.

It's not an explicit function of $t$. When you want to do chain rule, you have to pay attention on which variables are written explicitly, despite whether or not these variables are functions of yet another variable. For example take $F(t) = xy^2$ where $x = \sqrt{t}$ and $y=t-2$. If you want to calculate $dF/dt$, you can either first express $x$ and $y$ in terms of $t$ and then differentiate w.r.t. $t$ or let $F$ be expressed in $x$ and $y$ then use the chain rule
$$\frac{dF}{dt} = \frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt} .$$

I guess this problem should belong to multivariate calculus.
Thanks a lot! I now understand.

Svein
I do not agree with the formula shown in the OP (dimensional analysis again). The picture states that $\frac{d}{dt}x(t) = \Delta x$ which is obviously wrong. If x is distance and t is time, it tries to assert that velocity equals a (short) distance. The correct statement is $\frac{d}{dt}x(t) \cdot \Delta t= \Delta x$.
I can agree with the second line, but the third line is pure nonsense.

blue_leaf77
I do not agree with the formula shown in the OP (dimensional analysis again). The picture states that $\frac{d}{dt}x(t) = \Delta x$ which is obviously wrong. If x is distance and t is time, it tries to assert that velocity equals a (short) distance. The correct statement is $\frac{d}{dt}x(t) \cdot \Delta t= \Delta x$.
In the linked pdf file, the author for some reason sets $\Delta t = 1$.