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2

Here's a derivative-free explanation. For readers who are doing E&M at the college level, the other answers posted here are more comprehensive, but since the OP has stated a high-school knowledge with little math and physics knowledge, here's the primer: A vector is a quantity that, in order to be fully measured and described, needs to include both a ...

1

The two maxwell equations using divergence are $$div \vec{D} = \rho \\ div \vec{B} = 0$$ at least in differential form. In integral form they are maybe more clearer for you. They are $$\iint_{\partial V} \vec{D} \ d \vec{A} = \iiint_{V} \rho \ dV = Q(V) \\ \iint_{\partial V} \vec{B} \ d \vec{A} = 0$$ The first equation just means the electrical flux $D$ ...

1

Taking your (lack of) knowledge about differential geometry into account, this might be too hard to follow, but here it goes anyway: Let $u_1,\dots,u_n$ be some tangent vectors with base point $p$ and $\omega$ the volume form, ie $V = \omega_p(u_1,\dots,u_n)$ is the (possibly negative) volume of the parallelepiped spanned by these vectors. In case of three ...

4

Divergence is an operation that maps a vector field $\vec D(x,y,z)$ to a scalar field ${\rm div}\,\vec D(x,y,z)$. How do you calculate ${\rm div}\,\vec D(x,y,z)$? Either you follow the definitions using derivatives, which you can't if you don't know what a derivative is. Or you imagine the following: the vector field $\vec D(x,y,z)$ tells you about the ...

0

I think the answer to this is basically yes. You do have to be careful, because in general, you can't interpret inner products in relativity as measures of whether something is "orthogonal" to something else in the Euclidean sense. E.g., a lightlike vector has a zero inner product with itself. This is because the metric isn't the Euclidean metric. However, ...

1

To address the first question you can consider $$\nabla_av^b := v^b ,_a$$ since $$\nabla : \Gamma(E) \rightarrow \Gamma(E\otimes T^*M)$$ where E is any section(e.g. the vector field in question) and contracting it with the tangent vector field of the curve you get $$t^av^b,_a = 0$$ this is similar to contracting a vector field with a dual vector  ...

2

Just use the definition http://en.wikipedia.org/wiki/Divergence $\nabla\cdot F = r^{2n}+2nx^2r^{2(n-1)}+r^{2n}+2ny^2r^{2(n-1)}+r^{2n}+2nz^2r^{2(n-1)} = 3r^{2n}+2nr^{2n-1}$

-2

Nature exists withouth mathematics. Mathematics is just a language tool (a way of visualizing things about how nature behaves). Observations of nature show us that things try to flow to especific regions of space. By definitions, this regions are described as lower potentials (the more potential, the more capacity of doing things). Gradient of the potential ...

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