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I think that the Wolfram Demonstration Electric Field Lines Due to a Collection of Point Charges (available at in case of link rot) does a pretty good job.


Here is a way to do the manipulation. First I define a convenient way to set up vector fields, then I do the dot product and gradient. The main thing is that the application to the vector argument on the right is done element-wise, so that the natural Mathematica operation is Map (/@). To make the order of operations clearer, I also use two different ...

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