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In the case of a complex shape filiform distribution of current, are we allowed to determine the magnetic field created by sections of the distribution and then summing them up, like we do with a discrete distribution of charge when calculating the electric field?

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Yes, absolutely. In other words, the magnetic field also obeys the principle of superposition. This does break down if you consider back-reaction (i.e. the currents feel Lorentz forces from the magnetic fields); but it should always be fine for static systems.

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Thank you for your answer –  Carpediem Mar 16 '13 at 19:18
    
Does the direction of the current matter? For example if I have a semi- circle and I calculated its magnetic field when I moves clockwise. Is the magnetic field identical if I moves counterclockwise ? –  Carpediem Mar 16 '13 at 19:36
    
Yes, opposite current produces an opposite direction magnetic field. Generally one uses the right-hand rule –  zhermes Mar 16 '13 at 19:44
    
I am a little confused because I don't see how the direction of the current will involve in the Biot-Savart law.. –  Carpediem Mar 16 '13 at 19:46
    
In the Biot-Savart law, the current is a vector - so it includes direction. –  zhermes Mar 16 '13 at 19:48
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