Question

Calculate the line integral of the vector field F = (xy)i + (x − y)j along C, the triangle composed of three segments. C1 is the line segment from (3, 0) to (-3, 0). C2 is the line segment from (-3, 0) to (0, 3). C3 is the line segment from (0, 3) to (3, 0).

Answer 1

Do you know how to parametrize a line segment?

Answer 2

\[
\mathbf{r}(t) = P_1+(P_2-P_1)t,\,t\in[0,1]
\]

Answer 3

That parametrization goes from \(P_1\) to \(P_2\) when \(t\) goes from \(0\) to \(1\).

Answer 4

so, for each segemtn i parameterize?

Answer 5

Each line segment needs its own parametrization and thus its own integral.

Answer 6

so for c2, (-3,0) to (0,3) the parameterization would be

r(t) = -3 + t(6) ??

Answer 7

It's a vector.

Answer 8

yes so then i take the derivative and then dot it with the F(r(t)) right/

Answer 9

I was saying you had it wrong because you didn't even have a vector, for one.

Answer 10

But yes, in general you got the right idea.

Answer 11

hmm... im stumped on getting the vector, then...

Answer 12

wait would the vector be <3,3>

Answer 13

I'll do the first one, but after that it's on you.

Answer 14

I just need guidance is all, If i can see how one is done i can do the others...

Answer 15

"the line segment from (3, 0) to (-3, 0)"\[
\mathbf r(t)=\mathbf p_1 +(\mathbf p_2-\mathbf p_1)t
=
\begin{bmatrix}
3 \\ 0
\end{bmatrix}
+
\left(
\begin{bmatrix}
3 \\ 0
\end{bmatrix}
-
\begin{bmatrix}
-3 \\ 0
\end{bmatrix}
\right)t
=
\begin{bmatrix}
3 \\ 0
\end{bmatrix}
+
\begin{bmatrix}
6 \\ 0
\end{bmatrix}t
=
\begin{bmatrix}
3+6t \\ 0
\end{bmatrix}
\]

Answer 16

\[
r'(t) =
\begin{bmatrix}
6 \\ 0
\end{bmatrix}
\]\[
\mathbf F\circ \mathbf{r} (t) =
\begin{bmatrix}
xy \\ x-y
\end{bmatrix}
\circ
\begin{bmatrix}
3+6t \\ 0
\end{bmatrix}
=
\begin{bmatrix}
(3+6t)(0) \\ (3+6t)-(0)
\end{bmatrix}
=
\begin{bmatrix}
0 \\ 3+6t
\end{bmatrix}

\]\[
\int \mathbf F\cdot d\mathbf s=
\int_{t_0}^{t_1}[\mathbf F\circ \mathbf r(t)]\cdot \mathbf r'(t) \;dt
=
\int_0^1

\begin{bmatrix}
0 \\ 3+6t
\end{bmatrix}\cdot
\begin{bmatrix}
6 \\ 0
\end{bmatrix}dt
=\int_0^10\;dt = 0
\]