Roger's Equations Blog

Roger's Equations

This blog is all about science and technology (with occasional math thrown in for fun). The goal of this blog is to try and pass on the sense of excitement and wonder I feel when I read about these topics. I hope you enjoy the posts.

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Curl, Divergence, and Gradients.

Posted April 07, 2005 3:17 PM by Bayes

Anyone who becomes familair with the differntial form of Maxwell's equations has heard of the Curl, Divergence, Gradient, and Laplacian. Now no one beats mathworld.wolfram.com if you're interested in the mathematical definition of the terms above, but I find visualization helps me understand proofs using these mathematical objects a lot more. So here's how I picture Curl, Divergence, Gradients, and the Laplacian in my head. For Divergence, I picture the electric field lines coming out of a point charge. Divergence tells you how quickly or slowly the lines from the point charge are spreading out (diverging) in the space around that point charge. For Curl, I picture a wire with a constant current running through it. Hopefully you know that a current of this type will generate a constant magnetic field that curls around the wire following the right hand rule. The Curl tells you how tightly the field lines Curl around that wire. For Gradient, picture a hill with gradually decreasing slope till it crests. Clearly a ball placed at the top of the hill would accelerate slower than one placed on the steep side of a hill. The Gradient tells you the slope of the vector field. This is a good way to see why the grdient of a potential gives you a force. If force is mass times acceleration, and acceleration depends on the slope, so does force (which is causing the acceleration). Finally the Laplacian, which is the rate of the rate of change. This is tricky and the hardest to visualize for me, I welcome any suggestions anyone might have.

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