# Line integrals with respect to x and y

1. Jan 4, 2008

### dontdisturbmycircles

1. The problem statement, all variables and given/known data
I am having a bit of trouble relating the line integral of a function with respect to arc length with the line integrals with respect to x and y.

Last edited: Jan 4, 2008
2. Jan 4, 2008

### quasar987

It would be best to ask a specific question.

3. Jan 5, 2008

### foxjwill

Let's say we have a curve $$C$$ parameterized by a function $$\textbf{r}(t)=x(t)\textbf{i} + y(t)\textbf{j}$$. Differentiating with respect to $$t$$ we get

$$\frac{d\textbf{r}}{dt} = \frac{dx}{dt}\textbf{i} + \frac{dy}{dt}\textbf{j}.$$

Multiplying through by $$dt$$, we get

$$d\textbf{r} = dx\textbf{i} + dy\textbf{j}.$$

Plugging into the line integral, we get

$${\int_C \textbf{F} \cdot d\textbf{r} } = {\int_C (M\textbf{i}+N\textbf{j}) \cdot (dx\textbf{i} + dy\textbf{j})}={\int_C Mdx + Ndy}$$

where $$\textbf{F}=M\textbf{i}+N\textbf{j}.$$

4. Jan 5, 2008

### Gib Z

foxjwill is either a physicist or an engineer =]

5. Jan 5, 2008

### foxjwill

Neither, actually. ;) I'm a high school senior who's really into math. Is the way I formulated the answer the way a physicist or engineer would do it?

6. Jan 5, 2008

### Gib Z

"Multiplying through by dt" lol

7. Jan 5, 2008

### foxjwill

lol. Thought it was that. I got that terminology from my physics teacher.

8. Jan 5, 2008

### Gib Z

We'll, theres a lot of Physicists/Engineers on these forums, so I won't say anymore =]