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I'm afraid your statement that magnetic field at a point can have only on(e) "constant value" is not true, and you will have to learn a bit about special relativity to understand why. There is a classic thought experiment, which is to imagine the electromagnetic fields due to a charged particle at rest in what we'll call the stationary frame. Clearly, the ...

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Most of the answers posted here are incorrect. The Wikipedia page for the gradient says The gradient of $f$ is defined as the unique vector field whose dot product with any vector $v$ at each point $x$ is the directional derivative of $f$ along $v$. A look at Theodore Frankel's The Geometry of Physics confirms this. Other posters have said that the ...

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But magnetic field at that point can have only on "CONSTANT VALUE". Not true. If a particle is acted on by some combination of electrical and magnetic forces in one frame of reference, then in another frame of reference, it will be a different combination of electrical and magnetic forces. It's possible to have a force that's purely electrical in one ...

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There are two mathematical concepts which are both called vector. The first one, the vector from linear vector space is the basic "multicomponent object" which you seem to mainly talk about. The second notion of a vector is of a member of the so-called "tangent bundle" of a manifold. The second notion is the one which is defined equivalently with the ...

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Let me start with a tautology: Vectors are geometrical objects living on a vector space XD So far it says nothing, but we always have had the mental image of a vector as an arrow. A bit further into abstraction (still with or idea of an arrow representing a vector), one can find a set of transformations of vectors which preserves the properties of vectors, ...

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Mathematically, the idea of a vector is prior. You could define objects that fulfill all properties of a vector space without referring to components or anything. From the notion of a vector one can derive that there exist a maximum number of linearly independent vectors and any vector in your vector space can be represented uniquely by a linear combination ...

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