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So I am trying to solve this Differential system/ Complex analysis problem $$f(z) = iz$$ to sketch the Streamlines of the planar flow and need help understanding the logic or where I went wrong :

Step one:

$$ z=x+yi\\ f(z)=i(x+yi)\\ f(z)=xi-y $$

Step 2: From there I know that I can use the chain rule to derive:

$$ \frac{dy}{dt}=x \\ \frac{dx}{dt}=-y $$ Step 3: Which I can use the chain rule to combine

$$\frac{dy}{dx}=\frac{x}{-y} = xdx+ydy $$ and then

$$ \int{x}{dx}+\int{y}{dy}=0 $$ to get

Step 4: $${\frac{1}{2}x^{2}+\frac{1}{2}y^{2} + c}$$

Im not quite sure how the answer got these last two steps of $$\frac{1}{2}x^{2}+\frac{1}{2}y^{2} =\frac{1}{2}c^{2}$$ $$x^{2}+y^{2}=c^{2}$$ I think that they multiplied by two on last part and then squared the function. However im not sure how they integrated the constant the way that they did. So any clarity on this matter would be greatly appreciated?

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    The solution is $t \mapsto f(0) e^{it}$. That is, circles traversed in an anticlockwise direction. – 2017-02-25
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    Thats what I assumed based on how vectors from the complex plane move in: $$ z^1 \implies {z^4} $$ and just wanted to make sure that my thoughts on the use of the principle argument worked to map out the angles. – 2017-02-25
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    I don't follow your last remarks. If you just differentiate the function $t \mapsto {1 \over 2} (x^2+y^2)$ you will get zero from which the result follows. I don't follow your Step 3 about either. – 2017-02-25
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    Step three just implied that I identified the velocity field equation applied with Parametrics $$ f(z)=P(x,y)+iQ(x,y) \implies \\\frac{dx}{dt}={P(x,y)} \\\frac{dy}{dt}=Q(x,y) {}$$ Thus allows the family of solutions to the system of the first order differential equations denoted as streamlines of the planar flow associated with f(z) – 2017-02-25
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    I follow Step 2 which is what the previous comment is about, I do not follow Step 3 where you equate ${dy \over dx}$ with $xdx+ydy$. – 2017-02-25
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    So basically took the notion of the chain rule from elementary calculus and applied this to idea of parametric functions .Such that $$(\frac{dy}{dt}/\frac{dx}{dt})=\frac{dy}{dx} $$ then applied the Separation of variables technique to get $$\frac{dy}{dx}=\frac{x}{-y} \implies xdx+ydy$$ With the real parts equaling X and the imaginary parts equaling Y – 2017-02-25
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    I don't understand what your last implication means. My second remark shows how you can show that $|z|$ is constant on a solution trajectory. – 2017-02-25
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    I was just trying to highlight that you can set it up two ways. One you stated the parts with the reduced parametric variables, you could solve the ODE system through separation of parts. – 2017-02-25

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From $$ \int{x}{dx}+\int{y}{dy}=0 $$ we have: $$\frac{1}{2}x^{2}+\frac{1}{2}y^{2} =k$$ where $k$ is a number that can be written as $k=\frac{1}{2}c^{2}$ (the only important thing is that it is a constant), so $$ \frac{1}{2}x^{2}+\frac{1}{2}y^{2} =\frac{1}{2}c^{2} $$

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    So would you say $$ x^{2} +y^{2}= C$$ to be correct also? I was also hoping that someone could clarify the direction of the field. My suspicion is that it moves counterclockwise but not sure.... – 2017-02-25