What is the Meaning of the Dot Product in Integrals?

[akcyrus]

What does the following mean -
$$\int \textbf{F} \cdot d \textbf{r}$$
I know that its equal to work done. but i have problem in understanding it, what's the varibale. i am familiar with integrate cosxdx. what's is similar to dx in above equation. is it ds.
Now how can one get to this from above:
$$\frac{m}{2} \frac{d}{dt}(\textbf{v} \cdot \textbf{v}) dt$$

 

[HallsofIvy]

The variable is r, of course. You know that because the integral is with respect to r. As for getting form the first formula to the second, you can't. The integral gives the work done by force, F, so, under some circumstances, the energy added to the system by the force. That might all go into kinetic energy which is the second expression but you cannot, mathematically, derive one expression from the other.

 

[D H]

What does the following mean -
$$\int \textbf{F} \cdot d \textbf{r}$$
I know that its equal to work done. but i have problem in understanding it, what's the varibale.

The above is a path integral, or line integral. The force F acting on some object will change the object's velocity. If you can parameterize the velocity as a function of time, the above becomes

$$W = \int_C \mathbf F \cdot d \mathbf r = \int \mathbf F \cdot \mathbf v \;dt$$

For an object of constant mass,

$$\mathbf F = m\frac {d \mathbf v}{dt}$$

from which

$$W = \int\frac m 2 \frac d {dt}(\mathbf v \cdot \mathbf v) \,dt$$

immediately follows.