Question

how to calculate line integrals with respect to z

Answer 1

I know x and y, how different is z?

Answer 2

Just insert the definition and then calculate each complex component as a real integral.

Answer 3

ok, so for respect to x: integral from a to b of f(x(t),y(t)) times x'(t)dt so for z i just replace the x?

Answer 4

z, you mean ds?

Answer 5

\[\int\limits_{a}^{b}f(x(t),y(t))x'(t)dt\]

Answer 6

No, you need to take the tangent vector as a complex number. So \[x'(t) + iy'(t)\]

Answer 7

\[\int_a^b f(r) r' dt\] ?

Answer 8

So that would be \[\int_a^b{f(x(t),y(t))\cdot(x'(t)+iy'(t))dt}\]

Answer 9

for the parameterized function

Answer 10

oh, what does the i stand for?

Answer 11

don't worry about, it is not calc III

Answer 12

\[i = \sqrt{-1}\]

Answer 13

oh, that i lol

Answer 14

\[\int_a^b f(x,y).(x',y')dt\]

Answer 15

so that is the dot product of f(x,y) and (x', y')?

Answer 16

yes

Answer 17

but if you are just viewing \(ℂ\) just as \(ℝ^2\), you could say \[∫_a^b({\begin{matrix}f_1(x(t),y(t))x'(t)-f_2(x(t),y(t))y'(t)\\ f_1(x(t),y(t))y'(t)+f_2(x(t),y(t))x'(t)\end{matrix})dt}\]

Answer 18

ok, that makes sense. thanks guys! :)