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# week5 - 5 5.1 Vector and scalar fields scalar fields...

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5 Vector and scalar fields 5.1 scalar fields A “scalar field” is a fancy name for a function of space, i.e. it associates a real number with every position in some space, e.g. in 3D φ = φ ( x, y, z ). We’ve already encountered examples without calling them scalar fields, e.g. the temper- ature T ( x, y ) in a metal plate, or the electrostatic potential φ = φ ( x, y, z ). The gravitational potential is another, and it’s frequently convenient to think about potential “landscapes”, imagining that a set of hills is a kind of paradigm for a varying potential, since the height in this case scales with the potential mgh ( x, y ) itself. Formally, scalar is a word used to distinguish the field from a vector field. We can do this because a scalar field is invariant under the rotation of the coordinate system: φ 0 ( x 0 , y 0 , z 0 ) = φ ( x, y, z ) . (1) In other words, I may label the point on top of one of the hills by a different set of coordinates, but this doesn’t change the height I assign to it. This is in contrast to a vector field, where the values of the components do change in the new coordinate system, as we have discussed. 5.1.1 gradients of scalar fields If you’re standing on the hill somewhere, say not on the top, there’s one direction in xy space which gives you the direction of the fastest way down. This vector is

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week5 - 5 5.1 Vector and scalar fields scalar fields...

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