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I'm having trouble understanding what exactly the gradient of a scalar field represents. According to wikipedia and the textbooks I have it points in the direction of greatest increase and has a magnitude of greatest increase. This by itself seems fine. However, I have also been using it to find the normal of a surface and I don't understand how it can be both.

Also, do the properties( eg, what it represents) of the gradient change as you change dimensions and how does it work on a simple 2d graph?