I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic energy, still confuse me:
$$ \mathbf{F} \cdot \mathrm{d}\mathbf{x} = \mathbf{F} \cdot \mathbf{v} \mathrm{d} t = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} \cdot \mathbf{v} \mathrm{d} t = \mathbb{v} \cdot \mathrm{d} \mathbf{p} = \mathbf{v} \cdot \mathrm{d}(m \mathbf{v}).$$
Taken from here.
I want to understand the symbolic manipulation that often occurs when making meaningful integrations. I was taught that the ending 'dx' term simply signifies the variable to be integrated over. However, it is commonly used, for example, as a term to cancel things out. In general, I see a lot of symbolic manipulation with differential elements that I want to understand. Could you recommend something I could read to better understand this stuff?
Thank you.
It is actually possible to define these kind of concepts only when every statement is meaningful: the example you reported lacks of sense (or many passages are in the best case implied) and rigour although it is a very common way to present introductory physics. Just for the case you mentioned: given the position of a material point with respect to time as $\mathbf{r}=\mathbf{r}\left(t\right)$, the velocity vector is $\mathbf{v}=\mathbf{v}\left(t\right)=\dot{\mathbf{r}}\left(t\right)$. One can define the power of a force $\mathbf{F}=\mathbf{F}\left(\mathbf{r},\mathbf{v},t\right)$ acting on the material point as
$\mathbf{F}\cdot\mathbf{v}=\mathbf{F}\left(\mathbf{r},\mathbf{v},t\right)\cdot\mathbf{v}\left(t\right)$
The second law of dynamics gives $\mathbf{F}$ (intended as the resultant of all the forces acting on the mp) as $ \mathbf{F}\left(\mathbf{r},\mathbf{v},t\right)=\dot{\mathbf{p}}\left(t\right) $ If we assume that mp has constant mass $m\left(t\right)=m$ then $ \mathbf{F}\cdot\mathbf{v}\left(t\right)=m\dot{\mathbf{v}}\cdot\mathbf{v}=\frac{m}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathbf{v}\cdot\mathbf{v}\right)=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}m\left|\mathbf{v}\left(t\right)\right|^{2}\right) $ We can now integrate wrt time to obtain $ \int_{t_{1}}^{t_{2}}\mathrm{d}t\left(\mathbf{F}\left(\mathbf{r},\mathbf{v},t\right)\cdot\mathbf{v}\left(t\right)\right)=\frac{1}{2}m\left(\left|\mathbf{v}\left(t_{2}\right)\right|^{2}-\left|\mathbf{v}\left(t_{1}\right)\right|^{2}\right) $ which is the well known relationship between work of the force $\mathbf{F}$ and difference of kinetic energy $T\left(t\right)=\frac{1}{2}m\left|\mathbf{v}\left(t\right)\right|^{2}$. We can now proficiently restrict to positional forces (e.g. $\mathbf{F}=\mathbf{F}\left(\mathbf{r}\right)$) and to known trajectory $\gamma$ for mp (e.g. because of constraints); we can also choose a coordinate $s=s\left(t\right)$ along the curve to obtain $\int_{t_{1}}^{t_{2}}\mathrm{d}t\left(\mathbf{F}\left(\mathbf{r}\left(t\right)\right)\cdot\mathbf{v}\left(t\right)\right)=\int_{s_{1}=s\left(t_{1}\right)}^{s_{2}=s\left(t_{2}\right)}\mathrm{d}s\left(\mathbf{F}\left(s\right)\cdot\boldsymbol{\tau}\right)=\int_{s_{1}}^{s_{2}}\mathrm{d}s\left(F_{\parallel}\left(s\right)\right)$ where $\boldsymbol{\tau}\left(s\right)=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}$ is the tangent vector to the curve and the last result is the very common way to introduce "work of a force" but, as you can see, is only a particular case and formal manipulation as that of $\mathbf{F}\cdot\mathrm{d}\mathbf{x}$ are by the way only sensed inside integrals (with particular care, also!) as you were taught.