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I've seen in some textbooks, the line integral is defined as $\displaystyle \int_\gamma Pdx+Qdy$, where $\gamma$ is a path and P and Q are continuous functions.However, in other books the line integral is defined as $\displaystyle\int_a^b f([\gamma(t)])\gamma'(t)dt$, where $\gamma$ is a path and $f$ is a continuous function. I do not understand why these two definitions are equivalent?

Bumblebee
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  • Is $f$ a vector field in your second definition? – GReyes Nov 11 '21 at 00:07
  • @GReyes It says, $\gamma :[a, b]\to C$ is a smooth path and that $f$ is a complex-valued function which is defined and continuous on the trajectory of $\gamma$. Under these conditions we define the complex line integral of $f$ along $\gamma$, denoted $\int_\gamma f(z)dz =\int_a^b f([\gamma(t)]\gamma'(t)dt$. –  Nov 11 '21 at 00:12

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These are two different definitions. The second one involves complex multiplication so the complex structure. The first one is for real vector fields. They are related. If you set $f=P+iQ$ and $\gamma’(t)dt=dx+idy$ and perform the multiplication in your second definition you will see that both the real and imaginary parts of the resulting complex number are line integrals in the sense of your first definition.

GReyes
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  • Thanks! Sorry I did not say very clearly in the question that P and Q are also complex-valued. And I don't understand why $\gamma’(t)dt=dx+idy$? –  Nov 11 '21 at 00:37